## Browse Course Material

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- Dr. Casey Rodriguez

## Departments

- Mathematics

## As Taught In

- Mathematical Analysis

## Learning Resource Types

Real analysis, course description.

## Real Analysis

## Introduction [ edit | edit source ]

Real Analysis is a very straightforward subject, in that it is simply a nearly linear development of mathematical ideas you have come across throughout your story of mathematics. However, instead of relying on sometimes uncertain intuition (which we have all felt when we were solving a problem we did not understand), we will anchor it to a rigorous set of mathematical theorems. Throughout this book, we will begin to see that we do not need intuition to understand mathematics - we need a manual.

The overarching thesis of this book is how to define the real numbers axiomatically. How would that work? This book will read in this manner: we set down the properties which we think define the real numbers. We then prove from these properties - and these properties only - that the real numbers behave in the way which we have always imagined them to behave. We will then rework all our elementary theorems and facts we collected over our mathematical lives so that it all comes together, almost as if it always has been true before we analyzed it; that it was in fact rigorous all along - except that now we will know how it came to be.

Do not believe that once you have completed this book, mathematics is over. In other fields of academic study, there are glimpses of a strange realm of mathematics increasingly brought to the forefront of standard thought. After understanding this book, mathematics will now seem as though it is incomplete and lacking in concepts that maybe you have wondered before. In this book, we will provide glimpses of something more to mathematics than the real numbers and real analysis. After all, the mathematics we talk about here always seems to only involve one variable in a sea of numbers and operations and comparisons.

Note: A table of the math symbols used below and their definitions is available in the Appendix .

- Old Introduction
- Manual of Style – How to read this wikibook

A select list of chapters curated from other books are listed below. They should help develop your mathematical rigor that is a necessary mode of thought you will need in this book as well as in higher mathematics.

- The set theory notation and mathematical proofs, from the book Mathematical Proof
- The experience of working with calculus concepts, from the book Calculus

## The real numbers [ edit | edit source ]

This part of the book formalizes the various types of numbers we use in mathematics, up to the real numbers. This part focuses on the axiomatic properties (what we have defined to be true for the sake of analysis) of not just the numbers themselves but the arithmetic operations and the inequality comparators as well.

## Functions, Trigonometry, and Graphical Analysis [ edit | edit source ]

This part of the book formalizes the definition and usage of graphs, functions, as well as trigonometry. The most curious aspect of this section is its usage of graphics as a method of proof for certain properties, such as trigonometry. These methods of proof are mostly frowned upon (due to the inaccuracy and lack of rigorous definition when it comes to graphical proofs), but they are essential to derive the trigonometric relationships, as the analytical definition of the trigonometric functions will make using trigonometry too difficult—especially if they are described early on.

- Graphical Analysis
- Inverse Functions
- Trigonometric Functions, as Axioms
- Trigonometric Theorems, as Axioms

The following chapters will rigorously define the trigonometric functions. They should only be read after you have a good understanding of derivatives, integrals, and inverse functions.

- Trigonometric Functions, Defined
- Trigonometric Theorems, Defined

## Sequences and series (May 25, 2008) [ edit | edit source ]

This part of the book formalizes sequences of numbers bound by arithmetic, set, or logical relationships. This part focuses on concepts such as mathematical induction and the properties associated with sets that are enumerable with natural numbers as well as a limit set of integers.

## Metric Spaces [ edit | edit source ]

This part of the book formalizes the concept of distance in mathematics, and provides an introduction to the analysis of metric space.

- Metric Spaces
- Compact Sets
- Connected Sets
- Complete Sets
- Normed Linear Spaces

## Basic Topology of R n {\displaystyle \mathbb {R} ^{n}} (March 29, 2009) [ edit | edit source ]

This part of the book formalizes the concept of intervals in mathematics, and provides an introduction to topology.

- Open and Closed Sets
- Limit Points (Accumulation Points)
- Interior, Closure, Boundary

## Limits and Continuity [ edit | edit source ]

This part of the book formalizes the concept of limits and continuity and how they form a logical relationship between elementary and higher mathematics. This part focuses on the epsilon-delta definition, how proofs following epsilon-delta operate on, and the implications of limits. It also discusses other topics such as continuity, a special case of limits.

- Topological Continuity
- Total Variation
- The Exponential Function

## Differentiation [ edit | edit source ]

This part of the book formalizes differentiation and how they are used to describe the nature of functions. This part focuses on proving how derivatives study the nature of change of a function and how derivatives can provide properties to functions.

- Differentiation
- Applications of Derivatives
- Multivariable Derivatives

## Integration [ edit | edit source ]

This part of the book formalizes integration and how imagining what area means can yield many different forms of integration. This part focuses on proving how derivatives study the nature of change of a function and how derivatives can provide properties to functions.

- Darboux Integral – The method of integration this book defaults to.
- Riemann integration – The popular form of integration.
- Fundamental Theorem of Calculus
- Applications of Integration – The theorems and algebra that use integration
- Generalized Integration

## Sequences of Functions [ edit | edit source ]

- Pointwise Convergence
- Uniform Convergence

## Power Series [ edit | edit source ]

- Power Series

## Multivariate analysis [ edit | edit source ]

- Differentiation in R n
- Inverse Function Theorem

## Appendices [ edit | edit source ]

Here, you will find a list of unsorted chapters. Some of them listed here are highly advanced topics, while others are tools to aid you on your mathematical journey. Since this is the last heading for the wikibook, the necessary book endings are also located here.

- Dedekind's construction
- Landau notation
- List of Mathematical Symbols
- List of Theorems
- Bibliography

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## Navigation menu

## A Primer of Real Analysis

Dan Sloughter, Furman University

Copyright Year: 2009

Publisher: Dan Sloughter

Language: English

## Formats Available

Conditions of use.

Learn more about reviews.

Reviewed by Seonguk Kim, Assistant of Professor of Mathematics, DePauw University on 9/20/19

This book consists of all essential sections that students should know in the class, Analysis or Introduction of Real Analysis. First, in chapter 1, it has crucial prerequisite contents. Second, from chapter 2 to 8, the order of sections is... read more

Comprehensiveness rating: 4 see less

This book consists of all essential sections that students should know in the class, Analysis or Introduction of Real Analysis. First, in chapter 1, it has crucial prerequisite contents. Second, from chapter 2 to 8, the order of sections is reasonable and well-organized. But some instructors may skip chapters, 3, 4 and 8 because of the limit of time. Finally, I like the composition adding the exercises after the theorems because the student may be able to have ideas much easier.

Content Accuracy rating: 4

The content looks good and little error. Even though some notations are ambiguous and not easily understandable, overall is good.

Relevance/Longevity rating: 4

In the class, Analysis, students learn about the fundamental mathematical structures and concepts, and the related textbook also does not have any space adding the up to date contents. Nevertheless, I feel that this textbook provides a new view of the concepts.

Clarity rating: 5

All text is from the mathematics terminology that makes the writing lucid and readable.

Consistency rating: 5

In every chapter, it has used consistent letters and terminologies. Also, the composition is uniform using the order, 1. A brief description of the concepts, 2. Related definitions 3. Theorems 4. Examples 5. Exercise students should think about more.

Modularity rating: 5

The book breaks into separated sections, and each part is short and consists of readable and accessible text.

Organization/Structure/Flow rating: 3

The order of topics is in general. But it depends on the instructors. For example, I like to introduce the basic concepts, sets including cardinality (chapter 3), functions, logics before starting the sequences. Also, I have explained the idea, topology (chapter 4). So, in my opinion, it is better to organize the order of topics from fundamentals, including cardinality to more functions and to add the appendix, topology.

Interface rating: 5

This text has a lot of essential and useful figures and formulas. I believe the figures and graphs make students understand more easily.

Grammatical Errors rating: 5

It looks no grammatical errors. At least, I could not find them.

Cultural Relevance rating: 5

This textbook is for pure mathematics. So, I believe it has no inclusive issues about races, ethnicities, and backgrounds at all.

Overall, the textbook is very well-organized. I like the way how to organize the chapters. It is essential and nothing of unnecessary sections. Specifically, I like the composition adding the exercises after theorems and examples. If I use the book, I do not have to add more examples and suggest the students with the exercise problems. There are also some drawbacks to the book like ordering the topics. Nevertheless, I value this book in teaching the course Analysis.

## Table of Contents

1 Fundamentals

- 1.1 Sets and relations
- 1.2 Functions
- 1.3 Rational numbers
- 1.4 Real Numbers

2 Sequences and Series

- 2.1 Sequences
- 2.2 Infinite series

3 Cardinality

- 3.1 Binary representations
- 3.2 Countable and uncountable sets
- 3.3 Power sets

4 Topology of the Real Line

- 4.1 Intervals
- 4.2 Open sets
- 4.3 Closed sets
- 4.4 Compact Sets

5 Limits and Continuity

- 5.2 Monotonic functions
- 5.3 Limits to infinity and infinite limits
- 5.4 Continuous Functions

6 Derivatives

- 6.1 Best linear approximations
- 6.2 Derivatives
- 6.3 Mean Value Theorem
- 6.4 Discontinuities of derivatives
- 6.5 l'Hˆopital's rule
- 6.6 Taylor's Theorem

7 Integrals

- 7.1 Upper and lower integrals
- 7.2 Integrals
- 7.3 Integrability conditions
- 7.4 Properties of integrals
- 7.5 The Fundamental Theorem of Calculus
- 7.6 Taylor's theorem revisited
- 7.7 An improper integral

8 More Functions

- 8.1 The arctangent function
- 8.2 The tangent function
- 8.3 The sine and cosine Functions
- 8.4 The logarithm function
- 8.5 The exponential function

## Ancillary Material

About the book.

This is a short introduction to the fundamentals of real analysis. Although the prerequisites are few, I have written the text assuming the reader has the level of mathematical maturity of one who has completed the standard sequence of calculus courses, has had some exposure to the ideas of mathematical proof (including induction), and has an acquaintance with such basic ideas as equivalence relations and the elementary algebraic properties of the integers.

## About the Contributors

Dan Sloughter has been teaching Furman students since 1986, and became Professor of Mathematics in 1996. He previously served as an assistant professor at Santa Clara University from 1983-86, and at Boston College from 1981-83. He was also an instructor at Dartmouth College from 1979-81.

## Contribute to this Page

- © 2021

## Real Analysis: Foundations

- Sergei Ovchinnikov 0

## Department of Mathematics, San Francisco State University, San Francisco, USA

You can also search for this author in PubMed Google Scholar

Explores the foundations of real analysis using the framework of general ordered fields, demonstrating the multifaceted nature of the area

Illustrates the definitions and logical interrelations between core concepts of real analysis, using numerous examples and counterexamples

Presents the material in a self-contained manner, featuring three appendices and over 130 exercises

Includes supplementary material: sn.pub/extras

Part of the book series: Universitext (UTX)

12k Accesses

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## Table of contents (6 chapters)

Front matter, rational numbers.

Sergei Ovchinnikov

## Real Numbers

Continuous functions, differentiation, integration, infinite series, back matter.

This textbook explores the foundations of real analysis using the framework of general ordered fields, demonstrating the multifaceted nature of the area. Focusing on the logical structure of real analysis, the definitions and interrelations between core concepts are illustrated with the use of numerous examples and counterexamples. Readers will learn of the equivalence between various theorems and the completeness property of the underlying ordered field. These equivalences emphasize the fundamental role of real numbers in analysis.

Real Analysis: Foundations is ideal for students at the upper-undergraduate or beginning graduate level who are interested in the logical underpinnings of real analysis. With over 130 exercises, it is suitable for a one-semester course on elementary real analysis, as well as independent study.

- Real analysis ordered fields
- Real analysis rational numbers
- Real analysis textbook
- Real analysis foundations textbook
- Real analysis logic textbook
- Real analysis intro textbook
- Real analysis real numbers
- Real analysis completeness
- Real analysis logic
- Completeness axioms
- Continuous functions
- Infinite series
- intermediate value theorem
- Convex functions real analysis
- Absolute convergence
- Riemann integral
- Darboux integral
- Dedekind's construction real numbers
- Ordered fields

“I would happily recommend to someone already familiar with the topics in RA and who is looking for an overview of the theoretical bedrock of the subject. … this text is best used by someone already quite mathematically mature, looking to further their own understanding of the Real Analysis landscape through self-study.” (John Ross, MAA Reviews, January 6, 2023)

Book Title : Real Analysis: Foundations

Authors : Sergei Ovchinnikov

Series Title : Universitext

DOI : https://doi.org/10.1007/978-3-030-64701-8

Publisher : Springer Cham

eBook Packages : Mathematics and Statistics , Mathematics and Statistics (R0)

Copyright Information : The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021

Softcover ISBN : 978-3-030-64700-1 Published: 17 February 2021

eBook ISBN : 978-3-030-64701-8 Published: 16 February 2021

Series ISSN : 0172-5939

Series E-ISSN : 2191-6675

Edition Number : 1

Number of Pages : XII, 178

Number of Illustrations : 13 b/w illustrations

Topics : Real Functions , Measure and Integration , Sequences, Series, Summability , Mathematical Logic and Foundations , Analysis

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## List of real analysis topics

This is a list of articles that are considered real analysis topics.

## General topics

Sequences and series, convergence, derivatives, fundamental theorems, foundational topics, applied mathematical tools, infinite expressions, inequalities, orthogonal polynomials, field of sets, historical figures, related fields of analysis.

- Subsequential limit – the limit of some subsequence
- One-sided limit – either of the two limits of functions of real variables x, as x approaches a point from above or below
- Squeeze theorem – confirms the limit of a function via comparison with two other functions
- Big O notation – used to describe the limiting behavior of a function when the argument tends towards a particular value or infinity, usually in terms of simpler functions

( see also list of mathematical series )

- Generalized arithmetic progression – a sequence of numbers such that the difference between consecutive terms can be one of several possible constants
- Geometric progression – a sequence of numbers such that each consecutive term is found by multiplying the previous one by a fixed non-zero number
- Harmonic progression – a sequence formed by taking the reciprocals of the terms of an arithmetic progression
- Finite sequence – see sequence
- Infinite sequence – see sequence
- Divergent sequence – see limit of a sequence or divergent series
- Cauchy sequence – a sequence whose elements become arbitrarily close to each other as the sequence progresses
- Convergent series – a series whose sequence of partial sums converges
- Divergent series – a series whose sequence of partial sums diverges

- Binomial series – the Maclaurin series of the function f given by f ( x ) = (1 + x ) α
- Telescoping series
- Alternating series
- Divergent geometric series
- Harmonic series
- Fourier series
- Lambert series

## Summation methods

- Cesàro summation
- Euler summation
- Lambert summation
- Borel summation
- Summation by parts – transforms the summation of products of into other summations
- Cesàro mean
- Abel's summation formula

## More advanced topics

- Cauchy product –is the discrete convolution of two sequences
- Farey sequence – the sequence of completely reduced fractions between 0 and 1
- Oscillation – is the behaviour of a sequence of real numbers or a real-valued function, which does not converge, but also does not diverge to +∞ or −∞; and is also a quantitative measure for that.
- Indeterminate forms – algebraic expressions gained in the context of limits. The indeterminate forms include 0 0 , 0/0, 1 ∞ , ∞ − ∞, ∞/∞, 0 × ∞, and ∞ 0 .
- Pointwise convergence , Uniform convergence
- Absolute convergence , Conditional convergence
- Normal convergence
- Radius of convergence

## Convergence tests

- Integral test for convergence
- Cauchy's convergence test
- Direct comparison test
- Limit comparison test
- Alternating series test
- Dirichlet's test
- Stolz–Cesàro theorem – is a criterion for proving the convergence of a sequence
- Function of a real variable
- Real multivariable function
- Nowhere continuous function
- Weierstrass function
- Quasi-analytic function
- Non-analytic smooth function
- Flat function
- Bump function
- Differentiable function
- Square-integrable function , p-integrable function
- Bernstein's theorem on monotone functions – states that any real-valued function on the half-line [0, ∞) that is totally monotone is a mixture of exponential functions
- Inverse function
- Convex function , Concave function
- Singular function
- Weakly harmonic function
- Proper convex function
- Rational function
- Orthogonal function
- Implicit function theorem – allows relations to be converted to functions
- Measurable function
- Baire one star function
- Symmetric function
- Differential of a function
- Modulus of continuity
- Lipschitz continuity
- Semi-continuity
- Equicontinuous
- Absolute continuity
- Hölder condition – condition for Hölder continuity

## Distributions

- Dirac delta function
- Heaviside step function
- Hilbert transform
- Green's function
- Bounded variation
- Total variation
- Inflection point – found using second derivatives
- Directional derivative , Total derivative , Partial derivative

## Differentiation rules

- Linearity of differentiation
- Product rule
- Quotient rule
- Inverse function theorem – gives sufficient conditions for a function to be invertible in a neighborhood of a point in its domain, also gives a formula for the derivative of the inverse function

## Differentiation in geometry and topology

see also List of differential geometry topics

- Differentiable manifold
- Differentiable structure
- Submersion – a differentiable map between differentiable manifolds whose differential is everywhere surjective

(see also Lists of integrals )

- Fundamental theorem of calculus – a theorem of antiderivatives
- Multiple integral
- Iterated integral
- Cauchy principal value – method for assigning values to certain improper integrals
- Line integral
- Anderson's theorem – says that the integral of an integrable, symmetric, unimodal, non-negative function over an n -dimensional convex body ( K ) does not decrease if K is translated inwards towards the origin

## Integration and measure theory

see also List of integration and measure theory topics

- Riemann–Stieltjes integral
- Darboux integral
- Lebesgue integration
- Monotone convergence theorem – relates monotonicity with convergence
- Intermediate value theorem – states that for each value between the least upper bound and greatest lower bound of the image of a continuous function there is at least one point in its domain that the function maps to that value
- Rolle's theorem – essentially states that a differentiable function which attains equal values at two distinct points must have a point somewhere between them where the first derivative is zero
- Mean value theorem – that given an arc of a differentiable curve, there is at least one point on that arc at which the derivative of the curve is equal to the "average" derivative of the arc

- L'Hôpital's rule – uses derivatives to help evaluate limits involving indeterminate forms
- Abel's theorem – relates the limit of a power series to the sum of its coefficients
- Lagrange inversion theorem – gives the Taylor series of the inverse of an analytic function
- Darboux's theorem – states that all functions that result from the differentiation of other functions have the intermediate value property: the image of an interval is also an interval
- Heine–Borel theorem – sometimes used as the defining property of compactness

## Real numbers

- Natural number
- Rational number
- Irrational number
- Completeness of the real numbers
- Least-upper-bound property
- Extended real number line
- Dedekind cut

## Specific numbers

- Neighbourhood
- Derived set (mathematics)
- Completeness
- Partition of an interval
- Contraction mapping
- Fixed point – a point of a function that maps to itself
- Continued fraction
- Infinite products

See list of inequalities

- Triangle inequality
- Bernoulli's inequality
- Cauchy–Schwarz inequality
- Hölder's inequality
- Minkowski inequality
- Jensen's inequality
- Chebyshev's inequality
- Inequality of arithmetic and geometric means
- Generalized mean
- Arithmetic mean
- Geometric mean
- Harmonic mean
- Geometric–harmonic mean
- Arithmetic–geometric mean
- Weighted mean
- Quasi-arithmetic mean
- Hermite polynomials
- Laguerre polynomials
- Jacobi polynomials
- Gegenbauer polynomials
- Legendre polynomials
- Euclidean space
- Banach fixed point theorem – guarantees the existence and uniqueness of fixed points of certain self-maps of metric spaces, provides method to find them
- Complete metric space
- Sequence space
- Compact space
- Lebesgue measure
- Hausdorff measure
- Dominated convergence theorem – provides sufficient conditions under which two limit processes commute, namely Lebesgue integration and almost everywhere convergence of a sequence of functions.
- Sigma-algebra
- Michel Rolle (1652–1719)
- Brook Taylor (1685–1731)
- Leonhard Euler (1707–1783)
- Joseph-Louis Lagrange (1736–1813)
- Joseph Fourier (1768–1830)
- Bernard Bolzano (1781–1848)
- Augustin Cauchy (1789–1857)
- Niels Henrik Abel (1802–1829)
- Peter Gustav Lejeune Dirichlet (1805–1859)
- Karl Weierstrass (1815–1897)
- Eduard Heine (1821–1881)
- Pafnuty Chebyshev (1821–1894)
- Leopold Kronecker (1823–1891)
- Bernhard Riemann (1826–1866)
- Richard Dedekind (1831–1916)
- Rudolf Lipschitz (1832–1903)
- Camille Jordan (1838–1922)
- Jean Gaston Darboux (1842–1917)
- Georg Cantor (1845–1918)
- Ernesto Cesàro (1859–1906)
- Otto Hölder (1859–1937)
- Hermann Minkowski (1864–1909)
- Alfred Tauber (1866–1942)
- Felix Hausdorff (1868–1942)
- Émile Borel (1871–1956)
- Henri Lebesgue (1875–1941)
- Wacław Sierpiński (1882–1969)
- Johann Radon (1887–1956)
- Karl Menger (1902–1985)
- Asymptotic analysis – studies a method of describing limiting behaviour
- List of convexity topics
- List of harmonic analysis topics
- List of Fourier analysis topics
- List of Fourier-related transforms
- Complex analysis – studies the extension of real analysis to include complex numbers
- Functional analysis – studies vector spaces endowed with limit-related structures and the linear operators acting upon these spaces
- Nonstandard analysis – studies mathematical analysis using a rigorous treatment of infinitesimals .
- Calculus , the classical calculus of Newton and Leibniz .
- Non-standard calculus , a rigorous application of infinitesimals , in the sense of non-standard analysis , to the classical calculus of Newton and Leibniz.

## Related Research Articles

Complex analysis , traditionally known as the theory of functions of a complex variable , is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, applied mathematics; as well as in physics, including the branches of hydrodynamics, thermodynamics, quantum mechanics, and twistor theory. By extension, use of complex analysis also has applications in engineering fields such as nuclear, aerospace, mechanical and electrical engineering.

In mathematics, convolution is a mathematical operation on two functions that produces a third function that expresses how the shape of one is modified by the other. The term convolution refers to both the result function and to the process of computing it. It is defined as the integral of the product of the two functions after one is reflected about the y-axis and shifted. The choice of which function is reflected and shifted before the integral does not change the integral result. The integral is evaluated for all values of shift, producing the convolution function.

In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space C n . The existence of a complex derivative in a neighbourhood is a very strong condition: it implies that a holomorphic function is infinitely differentiable and locally equal to its own Taylor series ( analytic ). Holomorphic functions are the central objects of study in complex analysis.

In mathematics, a series is, roughly speaking, the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures through generating functions. In addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics, computer science, statistics and finance.

In mathematics, the mean value theorem states, roughly, that for a given planar arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant through its endpoints. It is one of the most important results in real analysis. This theorem is used to prove statements about a function on an interval starting from local hypotheses about derivatives at points of the interval.

In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include convergence, limits, continuity, smoothness, differentiability and integrability.

In mathematical physics, the Dirac delta distribution , also known as the unit impulse , is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line is equal to one.

Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions.

In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions converges uniformly to a limiting function on a set as the function domain if, given any arbitrarily small positive number , a number can be found such that each of the functions differs from by no more than at every point in . Described in an informal way, if converges to uniformly, then how quickly the functions approach is "uniform" throughout in the following sense: in order to guarantee that differs from by less than a chosen distance , we only need to make sure that is larger than or equal to a certain , which we can find without knowing the value of in advance. In other words, there exists a number that might depend on but is independent of , such that choosing will ensure that for all . In contrast, pointwise convergence of to merely guarantees that for any given in advance, we can find such that, for that particular , falls within of whenever .

In calculus, Taylor's theorem gives an approximation of a -times differentiable function around a given point by a polynomial of degree , called the -th-order Taylor polynomial . For a smooth function, the Taylor polynomial is the truncation at the order of the Taylor series of the function. The first-order Taylor polynomial is the linear approximation of the function, and the second-order Taylor polynomial is often referred to as the quadratic approximation . There are several versions of Taylor's theorem, some giving explicit estimates of the approximation error of the function by its Taylor polynomial.

In mathematical analysis, Lipschitz continuity , named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exists a real number such that, for every pair of points on the graph of this function, the absolute value of the slope of the line connecting them is not greater than this real number; the smallest such bound is called the Lipschitz constant of the function. For instance, every function that is defined on an interval and has bounded first derivative is Lipschitz continuous.

In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function where U is an open subset of that satisfies Laplace's equation, that is,

In mathematics, Cauchy's integral formula , named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary of the disk, and it provides integral formulas for all derivatives of a holomorphic function. Cauchy's formula shows that, in complex analysis, "differentiation is equivalent to integration": complex differentiation, like integration, behaves well under uniform limits – a result that does not hold in real analysis.

In calculus, Rolle's theorem or Rolle's lemma essentially states that any real-valued differentiable function that attains equal values at two distinct points must have at least one stationary point somewhere between them—that is, a point where the first derivative is zero. The theorem is named after Michel Rolle.

In mathematical analysis, Cesàro summation assigns values to some infinite sums that are not necessarily convergent in the usual sense. The Cesàro sum is defined as the limit, as n tends to infinity, of the sequence of arithmetic means of the first n partial sums of the series.

In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit.

In mathematics, the Riesz–Fischer theorem in real analysis is any of a number of closely related results concerning the properties of the space L 2 of square integrable functions. The theorem was proven independently in 1907 by Frigyes Riesz and Ernst Sigismund Fischer.

In mathematics, Hilbert spaces allow the methods of linear algebra and calculus to be generalized from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spaces. Formally, a Hilbert space is a vector space equipped with an inner product that induces a distance function for which the space is a complete metric space.

Most of the terms listed in Wikipedia glossaries are already defined and explained within Wikipedia itself. However, glossaries like this one are useful for looking up, comparing and reviewing large numbers of terms together. You can help enhance this page by adding new terms or writing definitions for existing ones.

## Real Analysis

Real analysis is a branch of mathematical analysis that analyses the behaviour of real numbers, sequences and series, and real functions. Convergence, limits, continuity, smoothness, differentiability, and integrability are some of the features of real-valued sequences and functions that real analysis explores. Complex analysis, on the other hand, is concerned with the study of complex numbers and their functions. In this article, let us discuss the brief introduction about the real analysis and the concepts involved with a complete explanation.

## Introduction to Real Analysis

As discussed above, real analysis is a branch of mathematics that was created to define the study of numbers and functions, as well as to analyze key concepts like limits and continuity. Calculus and its applications are based on these ideas. In a wide range of applications, real analysis has become a vital tool. Now, let us have a brief look at some of the important concepts covered under real analysis.

## Real Number System

The real number system (usually referred to as the reals) is first and foremost a set of numbers {a, b, c,…} on which the operations of addition and multiplication are defined such that every pair of real numbers does have a unique sum and product, with the properties listed below.

- Commutative Law: a+b = b + a and ab = ba
- Associative Law: (a + b) + c = a + (b + c) and (ab)c = a(bc)
- Distributive Law: a (b + c) = ab + ac
- For all a, there are unique real numbers 0 and 1, such that a+0 = a and a1=a.
- There is a real number -a for each a such that a + (-a) = 0, and if a ≠ 0 there is a real number 1/a for each a such that a(1/a) = 1.

Sequence: A sequence is defined as a function whose domain is the collection of positive integers.

(i.e) a n = a(n), where n = 1, 2, 3, ….

Assume that P n is the nth prime number, then

Convergence of Sequence: The sequence \(\begin{array}{l}\left\{ a_{n}\right\}_{n=1}^{\infty } \end{array} \) converges to the real number, say A, if and only if for each ∈ = 0, there should be a positive integer, say N, such that for all n≥N, and we have |a n – A| < ∈.

Infinite Series: An infinite series is defined as a pair \(\begin{array}{l}\left\{ \left\{ a_{n}\right\}_{n=1}^{\infty } , \left\{ S_{n}\right\}_{n=1}^{\infty } \right\}\end{array} \) , where \(\begin{array}{l}\left\{ a_{n}\right\}_{n=1}^{\infty } \end{array} \) represents the sequence of real numbers and \(\begin{array}{l}\sum_{k=1}^{\infty}a_{k}\end{array} \) for all n.

Here, a n is the nth term of the series and S n is the nth term of the partial sum of the series.

Convergence of Series: The converge of series states that, if \(\begin{array}{l}\sum_{n=1}^{\infty}a_{n}\end{array} \) converges, then \(\begin{array}{l}\left\{ S_{n}\right\}_{n=1}^{\infty } \end{array} \) converges.

Also, read: Sequence and Series

## Maxima and Minima

Maximum Value: The continuous function f(x) is considered to have a maximum value for x = a, if f(a) should be greater than any other values of f(x) that lies in the small neighbourhood of x = a.

Minimum Value: The continuous function f(x) is considered to have a minimum value at x = a, if f(a) should be smaller than any other values of f(x) that lie in the small neighbourhood of x = a.

Note: The tangent at maximum point or the minimum point of a curve should be parallel to x-axis.

## Metric Space

A metric space <x, P> is defined as a non-empty set X of points (elements) and P: X×X→R, such that x, y, z belongs to X.

- P(x, y) ≥ 0
- P(x, y) = 0, if and only if x = y
- P(x, y) = P(y, x)
- P(x, y) ≤ P(x, y) + P(x, y)

Here, the function “P” is called a metric.

## Solved Problems on Real Analysis

Determine the nth term of the sequence {0, 1, 0, 1, …}

Given sequence: {0, 1, 0, 1, …}

From the given sequence, we can write as follow:

The 1st term of sequence = 0 = a 1 = (1-1)/2 = 0

The 2nd term of sequence = 1 = a 2 = [1+(-1) 2 ]/2 = 1

The 3rd term of sequence = 0 = a 3 = [1+(-1) 3 ]/2 = 0

The 4th term of sequence = 1 = a 4 = [1+(-1) 4 ]/2 = 1

Hence, the nth term of sequence = a n = [1+(-1) n ]/2.

Evaluate using Green’s theorem, ∮ c (y- sinx dx + cos x dy), where c represents the triangle which is enclosed by the lines x = 0, x = π/2, xy = 2x and P = y-sinx and Q = cos x.

From the given conditions, we can write

∂Q/∂x = – sin x

By using the Green’s theorem, we can write

∮ c (y- sinx dx + cosx dy) = ∫∫ S (-1 – sin x) dx dy

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## Frequently Asked Questions on Real Analysis

What is real analysis in mathematics.

Real analysis is a branch of mathematics that studies how real numbers, sequences and series, and real functions behave.

## What is real analysis?

Real analysis is a discipline of mathematics that was developed to define the study of numbers and functions, as well as to investigate essential concepts such as limits and continuity. These concepts underpin calculus and its applications. Real analysis has become an incredible resource in a wide range of applications.

## Mention a few important topics that are covered under real analysis.

The important topics which are covered under real analysis are real number system, sequence and series, limits and continuity, integration, differentiation, Riemann integration, convergence and compactness, and so on.

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## List of real analysis topics

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Copyright © 2024 EPFL, all rights reserved

## List of real analysis topics

This is a list of articles that are considered real analysis topics.

General topics Limits

Limit of a sequence Subsequential limit – the limit of some subsequence Limit of a function (see List of limits for a list of limits of common functions) One-sided limit – either of the two limits of functions of real variables x, as x approaches a point from above or below Squeeze theorem – confirms the limit of a function via comparison with two other functions Big O notation – used to describe the limiting behavior of a function when the argument tends towards a particular value or infinity, usually in terms of simpler functions

Sequences and series

(see also list of mathematical series)

Arithmetic progression – a sequence of numbers such that the difference between the consecutive terms is constant Generalized arithmetic progression – a sequence of numbers such that the difference between consecutive terms can be one of several possible constants Geometric progression – a sequence of numbers such that each consecutive term is found by multiplying the previous one by a fixed non-zero number Harmonic progression – a sequence formed by taking the reciprocals of the terms of an arithmetic progression Finite sequence – see sequence Infinite sequence – see sequence Divergent sequence – see limit of a sequence or divergent series Convergent sequence – see limit of a sequence or convergent series Cauchy sequence – a sequence whose elements become arbitrarily close to each other as the sequence progresses Convergent series – a series whose sequence of partial sums converges Divergent series – a series whose sequence of partial sums diverges Power series – a series of the form \( f(x)=\sum _{n=0}^{\infty }a_{n}\left(x-c\right)^{n}=a_{0}+a_{1}(x-c)^{1}+a_{2}(x-c)^{2}+a_{3}(x-c)^{3}+\cdots \) Taylor series – a series of the form \( f(a)+{\frac {f'(a)}{1!}}(x-a)+{\frac {f''(a)}{2!}}(x-a)^{2}+{\frac {f^{(3)}(a)}{3!}}(x-a)^{3}+\cdots \). Maclaurin series – see Taylor series Binomial series – the Maclaurin series of the function f given by f(x) = (1 + x) α Telescoping series Alternating series Geometric series Divergent geometric series Harmonic series Fourier series Lambert series

Summation methods

Cesàro summation Euler summation Lambert summation Borel summation Summation by parts – transforms the summation of products of into other summations Cesàro mean Abel's summation formula

More advanced topics

Convolution Cauchy product –is the discrete convolution of two sequences Farey sequence – the sequence of completely reduced fractions between 0 and 1 Oscillation – is the behaviour of a sequence of real numbers or a real-valued function, which does not converge, but also does not diverge to +∞ or −∞; and is also a quantitative measure for that. Indeterminate forms – algebraic expressions gained in the context of limits. The indeterminate forms include 00, 0/0, 1∞, ∞ − ∞, ∞/∞, 0 × ∞, and ∞0.

Convergence

Pointwise convergence, Uniform convergence Absolute convergence, Conditional convergence Normal convergence Radius of convergence

Convergence tests

Integral test for convergence Cauchy's convergence test Ratio test Direct comparison test Limit comparison test Root test Alternating series test Dirichlet's test Stolz–Cesàro theorem – is a criterion for proving the convergence of a sequence

Function of a real variable Real multivariable function Continuous function Nowhere continuous function Weierstrass function Smooth function Analytic function Quasi-analytic function Non-analytic smooth function Flat function Bump function Differentiable function Integrable function Square-integrable function, p-integrable function Monotonic function Bernstein's theorem on monotone functions – states that any real-valued function on the half-line [0, ∞) that is totally monotone is a mixture of exponential functions Inverse function Convex function, Concave function Singular function Harmonic function Weakly harmonic function Proper convex function Rational function Orthogonal function Implicit and explicit functions Implicit function theorem – allows relations to be converted to functions Measurable function Baire one star function Symmetric function Domain Codomain Image Support Differential of a function

Uniform continuity Modulus of continuity Lipschitz continuity Semi-continuity Equicontinuous Absolute continuity Hölder condition – condition for Hölder continuity

Distributions

Dirac delta function Heaviside step function Hilbert transform Green's function

Bounded variation Total variation

Derivatives

Second derivative Inflection point – found using second derivatives Directional derivative, Total derivative, Partial derivative

Differentiation rules

Linearity of differentiation Product rule Quotient rule Chain rule Inverse function theorem – gives sufficient conditions for a function to be invertible in a neighborhood of a point in its domain, also gives a formula for the derivative of the inverse function

Differentiation in geometry and topology

see also List of differential geometry topics

Differentiable manifold Differentiable structure Submersion – a differentiable map between differentiable manifolds whose differential is everywhere surjective

(see also Lists of integrals)

Antiderivative Fundamental theorem of calculus – a theorem of antiderivatives Multiple integral Iterated integral Improper integral Cauchy principal value – method for assigning values to certain improper integrals Line integral Anderson's theorem – says that the integral of an integrable, symmetric, unimodal, non-negative function over an n-dimensional convex body (K) does not decrease if K is translated inwards towards the origin

Integration and measure theory

see also List of integration and measure theory topics

Riemann integral, Riemann sum Riemann–Stieltjes integral Darboux integral Lebesgue integration

Fundamental theorems

Monotone convergence theorem – relates monotonicity with convergence Intermediate value theorem – states that for each value between the least upper bound and greatest lower bound of the image of a continuous function there is at least one point in its domain that the function maps to that value Rolle's theorem – essentially states that a differentiable function which attains equal values at two distinct points must have a point somewhere between them where the first derivative is zero Mean value theorem – that given an arc of a differentiable curve, there is at least one point on that arc at which the derivative of the curve is equal to the "average" derivative of the arc Taylor's theorem – gives an approximation of a k {\displaystyle k} k times differentiable function around a given point by a k {\displaystyle k} k-th order Taylor-polynomial. L'Hôpital's rule – uses derivatives to help evaluate limits involving indeterminate forms Abel's theorem – relates the limit of a power series to the sum of its coefficients Lagrange inversion theorem – gives the Taylor series of the inverse of an analytic function Darboux's theorem – states that all functions that result from the differentiation of other functions have the intermediate value property: the image of an interval is also an interval Heine–Borel theorem – sometimes used as the defining property of compactness Bolzano–Weierstrass theorem – states that each bounded sequence in R n {\displaystyle \mathbb {R} ^{n}} \mathbb {R} ^{n} has a convergent subsequence Extreme value theorem - states that if a function f {\displaystyle f} f is continuous in the closed and bounded interval [ a , b ] {\displaystyle [a,b]} [a,b], then it must attain a maximum and a minimum

Foundational topics Numbers Real numbers

Construction of the real numbers Natural number Integer Rational number Irrational number Completeness of the real numbers Least-upper-bound property Real line Extended real number line Dedekind cut

Specific numbers

0 1 0.999... Infinity

Open set Neighbourhood Cantor set Derived set (mathematics) Completeness Limit superior and limit inferior Supremum Infimum Interval Partition of an interval

Contraction mapping Metric map Fixed point – a point of a function that maps to itself

Applied mathematical tools Infinite expressions

Continued fraction Series Infinite products

Inequalities

See list of inequalities

Triangle inequality Bernoulli's inequality Cauchy–Schwarz inequality Hölder's inequality Minkowski inequality Jensen's inequality Chebyshev's inequality Inequality of arithmetic and geometric means

Generalized mean Pythagorean means Arithmetic mean Geometric mean Harmonic mean Geometric–harmonic mean Arithmetic–geometric mean Weighted mean Quasi-arithmetic mean

Orthogonal polynomials

Classical orthogonal polynomials Hermite polynomials Laguerre polynomials Jacobi polynomials Gegenbauer polynomials Legendre polynomials

Euclidean space Metric space Banach fixed point theorem – guarantees the existence and uniqueness of fixed points of certain self-maps of metric spaces, provides method to find them Complete metric space Topological space Function space Sequence space Compact space

Lebesgue measure Outer measure Hausdorff measure Dominated convergence theorem – provides sufficient conditions under which two limit processes commute, namely Lebesgue integration and almost everywhere convergence of a sequence of functions.

Field of sets

Sigma-algebra

Historical figures

Michel Rolle (1652–1719) Brook Taylor (1685–1731) Leonhard Euler (1707–1783) Joseph-Louis Lagrange (1736–1813) Joseph Fourier (1768–1830) Bernard Bolzano (1781–1848) Augustin Cauchy (1789–1857) Niels Henrik Abel (1802–1829) Peter Gustav Lejeune Dirichlet (1805–1859) Karl Weierstrass (1815–1897) Eduard Heine (1821–1881) Pafnuty Chebyshev (1821–1894) Leopold Kronecker (1823–1891) Bernhard Riemann (1826–1866) Richard Dedekind (1831–1916) Rudolf Lipschitz (1832–1903) Camille Jordan (1838–1922) Jean Gaston Darboux (1842–1917) Georg Cantor (1845–1918) Ernesto Cesàro (1859–1906) Otto Hölder (1859–1937) Hermann Minkowski (1864–1909) Alfred Tauber (1866–1942) Felix Hausdorff (1868–1942) Émile Borel (1871–1956) Henri Lebesgue (1875–1941) Wacław Sierpiński (1882–1969) Johann Radon (1887–1956) Karl Menger (1902–1985)

Related fields of analysis

Asymptotic analysis – studies a method of describing limiting behaviour Convex analysis – studies the properties of convex functions and convex sets List of convexity topics Harmonic analysis – studies the representation of functions or signals as superpositions of basic waves List of harmonic analysis topics Fourier analysis – studies Fourier series and Fourier transforms List of Fourier analysis topics List of Fourier-related transforms Complex analysis – studies the extension of real analysis to include complex numbers Functional analysis – studies vector spaces endowed with limit-related structures and the linear operators acting upon these spaces Nonstandard analysis – studies mathematical analysis using a rigorous treatment of infinitesimals.

Calculus, the classical calculus of Newton and Leibniz. Non-standard calculus, a rigorous application of infinitesimals, in the sense of non-standard analysis, to the classical calculus of Newton and Leibniz.

Undergraduate Texts in Mathematics

Graduate Texts in Mathematics

Graduate Studies in Mathematics

Mathematics Encyclopedia

Hellenica World - Scientific Library

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## List of real analysis topics

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This is a list of articles that are considered real analysis topics.

- 1.2.1 Summation methods
- 1.2.2 More advanced topics
- 1.3.1 Convergence tests
- 1.4.1 Continuity
- 1.4.2 Distributions
- 1.4.3 Variation
- 1.5.1 Differentiation rules
- 1.5.2 Differentiation in geometry and topology
- 1.6.1 Integration and measure theory
- 2 Fundamental theorems
- 3.1.1 Real numbers
- 3.1.2 Specific numbers
- 4.1 Infinite expressions
- 4.2 Inequalities
- 4.4 Orthogonal polynomials
- 4.6 Measures
- 4.7 Field of sets
- 5 Historical figures
- 6 Related fields of analysis

## General topics

- Subsequential limit – the limit of some subsequence
- One-sided limit – either of the two limits of functions of real variables x, as x approaches a point from above or below
- Squeeze theorem – confirms the limit of a function via comparison with two other functions
- Big O notation – used to describe the limiting behavior of a function when the argument tends towards a particular value or infinity, usually in terms of simpler functions

## Sequences and series

( see also list of mathematical series )

- Generalized arithmetic progression – a sequence of numbers such that the difference between consecutive terms can be one of several possible constants
- Geometric progression – a sequence of numbers such that each consecutive term is found by multiplying the previous one by a fixed non-zero number
- Harmonic progression – a sequence formed by taking the reciprocals of the terms of an arithmetic progression
- Finite sequence – see sequence
- Infinite sequence – see sequence
- Divergent sequence – see limit of a sequence or divergent series
- Cauchy sequence – a sequence whose elements become arbitrarily close to each other as the sequence progresses
- Convergent series – a series whose sequence of partial sums converges
- Divergent series – a series whose sequence of partial sums diverges
- Binomial series – the Maclaurin series of the function f given by f ( x ) = (1 + x ) α
- Telescoping series
- Alternating series
- Divergent geometric series
- Harmonic series
- Fourier series
- Lambert series

## Summation methods

- Cesàro summation
- Euler summation
- Lambert summation
- Borel summation
- Summation by parts – transforms the summation of products of into other summations
- Cesàro mean
- Abel's summation formula

## More advanced topics

- Cauchy product –is the discrete convolution of two sequences
- Farey sequence – the sequence of completely reduced fractions between 0 and 1
- Oscillation – is the behaviour of a sequence of real numbers or a real-valued function, which does not converge, but also does not diverge to +∞ or −∞; and is also a quantitative measure for that.
- Indeterminate forms – algebraic expressions gained in the context of limits. The indeterminate forms include 0 0 , 0/0, 1 ∞ , ∞ − ∞, ∞/∞, 0 × ∞, and ∞ 0 .

## Convergence

- Pointwise convergence , Uniform convergence
- Absolute convergence , Conditional convergence
- Normal convergence
- Radius of convergence

## Convergence tests

- Integral test for convergence
- Cauchy's convergence test
- Direct comparison test
- Limit comparison test
- Alternating series test
- Dirichlet's test
- Stolz–Cesàro theorem – is a criterion for proving the convergence of a sequence
- Function of a real variable
- Real multivariable function
- Nowhere continuous function
- Weierstrass function
- Quasi-analytic function
- Non-analytic smooth function
- Flat function
- Bump function
- Differentiable function
- Square-integrable function , p-integrable function
- Bernstein's theorem on monotone functions – states that any real-valued function on the half-line [0, ∞) that is totally monotone is a mixture of exponential functions
- Inverse function
- Convex function , Concave function
- Singular function
- Weakly harmonic function
- Proper convex function
- Rational function
- Orthogonal function
- Implicit function theorem – allows relations to be converted to functions
- Measurable function
- Baire one star function
- Symmetric function
- Differential of a function
- Modulus of continuity
- Lipschitz continuity
- Semi-continuity
- Equicontinuous
- Absolute continuity
- Hölder condition – condition for Hölder continuity

## Distributions

- Dirac delta function
- Heaviside step function
- Hilbert transform
- Green's function
- Bounded variation
- Total variation

## Derivatives

- Inflection point – found using second derivatives
- Directional derivative , Total derivative , Partial derivative

## Differentiation rules

- Linearity of differentiation
- Product rule
- Quotient rule
- Inverse function theorem – gives sufficient conditions for a function to be invertible in a neighborhood of a point in its domain, also gives a formula for the derivative of the inverse function

## Differentiation in geometry and topology

see also List of differential geometry topics

- Differentiable manifold
- Differentiable structure
- Submersion – a differentiable map between differentiable manifolds whose differential is everywhere surjective

(see also Lists of integrals )

- Fundamental theorem of calculus – a theorem of antiderivatives
- Multiple integral
- Iterated integral
- Cauchy principal value – method for assigning values to certain improper integrals
- Line integral
- Anderson's theorem – says that the integral of an integrable, symmetric, unimodal, non-negative function over an n -dimensional convex body ( K ) does not decrease if K is translated inwards towards the origin

## Integration and measure theory

see also List of integration and measure theory topics

- Riemann–Stieltjes integral
- Darboux integral
- Lebesgue integration

## Fundamental theorems

- Monotone convergence theorem – relates monotonicity with convergence
- Intermediate value theorem – states that for each value between the least upper bound and greatest lower bound of the image of a continuous function there is at least one point in its domain that the function maps to that value
- Rolle's theorem – essentially states that a differentiable function which attains equal values at two distinct points must have a point somewhere between them where the first derivative is zero
- Mean value theorem – that given an arc of a differentiable curve, there is at least one point on that arc at which the derivative of the curve is equal to the "average" derivative of the arc
- Taylor's theorem – gives an approximation of a [math]\displaystyle{ k }[/math] times differentiable function around a given point by a [math]\displaystyle{ k }[/math] -th order Taylor-polynomial.
- L'Hôpital's rule – uses derivatives to help evaluate limits involving indeterminate forms
- Abel's theorem – relates the limit of a power series to the sum of its coefficients
- Lagrange inversion theorem – gives the Taylor series of the inverse of an analytic function
- Darboux's theorem – states that all functions that result from the differentiation of other functions have the intermediate value property: the image of an interval is also an interval
- Heine–Borel theorem – sometimes used as the defining property of compactness
- Bolzano–Weierstrass theorem – states that each bounded sequence in [math]\displaystyle{ \mathbb{R}^{n} }[/math] has a convergent subsequence
- Extreme value theorem - states that if a function [math]\displaystyle{ f }[/math] is continuous in the closed and bounded interval [math]\displaystyle{ [a,b] }[/math] , then it must attain a maximum and a minimum

## Foundational topics

Real numbers.

- Natural number
- Rational number
- Irrational number
- Completeness of the real numbers
- Least-upper-bound property
- Extended real number line
- Dedekind cut

## Specific numbers

- Neighbourhood
- Derived set (mathematics)
- Completeness
- Partition of an interval
- Contraction mapping
- Fixed point – a point of a function that maps to itself

## Applied mathematical tools

Infinite expressions.

- Continued fraction
- Infinite products

## Inequalities

See list of inequalities

- Triangle inequality
- Bernoulli's inequality
- Cauchy–Schwarz inequality
- Hölder's inequality
- Minkowski inequality
- Jensen's inequality
- Chebyshev's inequality
- Inequality of arithmetic and geometric means
- Generalized mean
- Arithmetic mean
- Geometric mean
- Harmonic mean
- Geometric–harmonic mean
- Arithmetic–geometric mean
- Weighted mean
- Quasi-arithmetic mean

## Orthogonal polynomials

- Hermite polynomials
- Laguerre polynomials
- Jacobi polynomials
- Gegenbauer polynomials
- Legendre polynomials
- Euclidean space
- Banach fixed point theorem – guarantees the existence and uniqueness of fixed points of certain self-maps of metric spaces, provides method to find them
- Complete metric space
- Sequence space
- Compact space
- Lebesgue measure
- Hausdorff measure
- Dominated convergence theorem – provides sufficient conditions under which two limit processes commute, namely Lebesgue integration and almost everywhere convergence of a sequence of functions.

## Field of sets

- Sigma-algebra

## Historical figures

- Michel Rolle (1652–1719)
- Brook Taylor (1685–1731)
- Leonhard Euler (1707–1783)
- Joseph-Louis Lagrange (1736–1813)
- Joseph Fourier (1768–1830)
- Bernard Bolzano (1781–1848)
- Augustin Cauchy (1789–1857)
- Niels Henrik Abel (1802–1829)
- Peter Gustav Lejeune Dirichlet (1805–1859)
- Karl Weierstrass (1815–1897)
- Eduard Heine (1821–1881)
- Pafnuty Chebyshev (1821–1894)
- Leopold Kronecker (1823–1891)
- Bernhard Riemann (1826–1866)
- Richard Dedekind (1831–1916)
- Rudolf Lipschitz (1832–1903)
- Camille Jordan (1838–1922)
- Jean Gaston Darboux (1842–1917)
- Georg Cantor (1845–1918)
- Ernesto Cesàro (1859–1906)
- Otto Hölder (1859–1937)
- Hermann Minkowski (1864–1909)
- Alfred Tauber (1866–1942)
- Felix Hausdorff (1868–1942)
- Émile Borel (1871–1956)
- Henri Lebesgue (1875–1941)
- Wacław Sierpiński (1882–1969)
- Johann Radon (1887–1956)
- Karl Menger (1902–1985)

## Related fields of analysis

- Asymptotic analysis – studies a method of describing limiting behaviour
- List of convexity topics
- List of harmonic analysis topics
- List of Fourier analysis topics
- List of Fourier-related transforms
- Complex analysis – studies the extension of real analysis to include complex numbers
- Functional analysis – studies vector spaces endowed with limit-related structures and the linear operators acting upon these spaces
- Nonstandard analysis – studies mathematical analysis using a rigorous treatment of infinitesimals.
- Calculus , the classical calculus of Newton and Leibniz .
- Non-standard calculus , a rigorous application of infinitesimals, in the sense of non-standard analysis , to the classical calculus of Newton and Leibniz.
- Real analysis
- Outlines of mathematics and logic
- Mathematics-related lists

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## Sets and Numbers

This chapter covers set theory. The topics include set algebra, relations, orderings and mappings, countability and sequences, real numbers, sequences and limits, and set classes including monotone classes, rings, fields, and sigma fields. The final section introduces the basic ideas of real analysis including Euclidean distance, sets of the real line, coverings, and compactness.

International Conference on Mathematical and Statistical Sciences (ICMSS) 2021 Study Program of Mathematics and the Study Program of Statistics Faculty of Mathematics and Natural Sciences Universitas Lambung Mangkurat Banjarbaru - Indonesia, 15 - 16 September 2021 The International Conference on Mathematical and Statistical Sciences (ICMSS) 2021 was organized through a collaboration between the Study Program of Mathematics and the Study Program of Statistics, Faculty of Mathematics and Natural Sciences - Universitas Lambung Mangkurat (ULM). The theme raised was “Mathematical and Statistical Sciences in Multidisciplinary Research”, with the aims are to acknowledge, learn, share, and transfer the results of scientific knowledge and research among academia and practitioners who have used or implemented Mathematical and Statistical Sciences to solve real-world problems and improve the quality of life. The scopes of our conference are Mathematical modeling, Artificial intelligence, Mathematical physics, Algebra and its applications, Statistics and its applications, Computational fluid dynamics, Data mining and its applications, Dynamical nonlinear systems, Mathematics Educations, Financial mathematics, Mathematical biology, Numerical methods and analysis, Operation research and optimizations, and Real analysis. On behalf of the committee, we would like to thank the Rector of Universitas Lambung Mangkurat, the Dean of Faculty of Mathematics and Natural Sciences, Coordinator of the Study Program of Mathematics and Coordinator of the Study Program of Statistics, advisory board, steering committee, all committee members, reviewers, presenters, and participants. We also would thank the Indonesian Mathematical Society (IndoMS), The Indonesian Algebra Society (IAS), and The Forum Pendidikan Tinggi Statistika (Forstat). Special thanks are also given to the Journal of Physics: Conference Series. We, on behalf of the ICMSS 2021 committee, would like to thank all parties for their participation in supporting this publication. We hope to see you all at the next conference. Kind regards, Dr. Muhammad Ahsar Karim Chair of the ICMSS 2021 List of Organizing Committees, Photographs and Peer review statement are available in this pdf.

## Roadmap to glory: scaffolding real analysis for deeper learning

Real analysis, analisis kemampuan berpikir tingkat tinggi mahasiswa dalam mengkonstruksi representasi biner bilangan real.

Higher-order thinking skills (HOTS) are needed to determine the student's ability to construct an answer. In this study, researchers analyzed the higher-order thinking skills of students of the Mathematics Education Study Program in constructing one of the test answers, namely constructing a binary representation of real numbers in the Introduction to Real Analysis course. Fifty-two students taking the Introduction to Real Analysis course in the odd semester 2020/2021 are the subjects of this research. Data was collected using a test that was analyzed based on the indicator of higher-order thinking ability created (C6). It was revealed that the students' higher-order thinking skills were in the sufficient category. This means that most students have not been able to construct and analyze information into the right strategy. The results of this study are expected to be a reference for the lecture process where students are familiarized with giving HOTS-oriented questions during exams and practice questions for lectures to help develop higher-order thinking skills.Keywords: Bloom's taxonomy-C6; higher order thinking skill; binary representation Kemampuan berpikir tingkat tinggi diperlukan untuk mengetahui kemampuan mahasiswa mengkonstruksi suatu jawaban. Pada studi ini, peneliti menganalisis kemampuan berpikir tingkat tinggi mahasiswa Program Studi Pendidikan Matematika dalam mengkonstruksi salah satu jawaban tes yaitu mengkonstruksi representasi biner bilangan real pada mata kuliah Pengantar Analisis Real. Lima puluh dua mahasiswa yang sedang mengambil mata kuliah Pengantar Analisis Real pada semester ganjil 2020/2021 sebagai subjek penelitian ini. Data dikumpulkan dengan tes yang dianalisis berdasarkan indikator kemampuan berpikir tingkat tinggi create (C6). Terungkap bahwa kemampuan berpikir tingkat tinggi mahasiswa berada pada kategori cukup. Ini berarti sebagian besar mahasiswa belum mampu mengkonstruksi dan menganalisis informasi menjadi strategi yang tepat. Hasil penelitian ini diharapkan menjadi acuan untuk proses perkuliahan dimana mahasiswa dibiasakan dengan pemberian soal yang berorientasi HOTS baik itu pada saat ujian maupun latihan-latihan soal perkuliahan untuk membantu mengembangkan kemampuan berpikir tingkat tinggi.Kata Kunci: taksonomi Bloom-C6; kemampuan berpikir tingkat tinggi; representasi biner

## On two kinds of the reverse half-discrete Mulholland-type inequalities involving higher-order derivative function

AbstractBy means of the weight functions, Hermite–Hadamard’s inequality, and the techniques of real analysis, a new more accurate reverse half-discrete Mulholland-type inequality involving one higher-order derivative function is given. The equivalent statements of the best possible constant factor related to a few parameters, the equivalent forms, and several particular inequalities are provided. Another kind of the reverses is also considered.

## Real Analysis, Harmonic Analysis and Applications

Ε and δ real analysis, mathematical-analytical thinking skills: the impacts and interactions of open-ended learning method & self-awareness (its application on bilingual test instruments).

Analytical thinking is a skill to unite the initial process, plan solutions, produce solutions, and conclude something to produce conclusions or correct answers. This research aims to 1) determine whether there are differences in students' mathematical, analytical thinking skills between classes that use the Open-ended learning method and classes that use the lecturing method, 2) to find out whether there are mathematical, analytical thinking skills differences between students with high, moderate, and low self-awareness criteria, and 3) to find out whether there is an interaction between Open-ended learning method and self-awareness toward students' mathematical-analytical thinking skills. This research employs a quasi-experimental design. Based on the data and data analysis, this research is mixed-method research, and the design used in this research is the posttest control group design. This research was conducted on students who have studied the Real Analysis Courses. Based on the results of hypothesis testing, it was found out that, first, there are differences in students' mathematical-analytical thinking skills between the class that uses the Open-ended learning method and the class that uses the lecturing method. Second, there are mathematical-analytical thinking skills differences between high, moderate, and low self-awareness criteria. Third, there is no interaction between the Open-ended learning method with self-awareness of students' mathematical-analytical thinking skills.

## Equivalent Properties of Two Kinds of Hardy-Type Integral Inequalities

In this paper, using weight functions as well as employing various techniques from real analysis, we establish a few equivalent conditions of two kinds of Hardy-type integral inequalities with nonhomogeneous kernel. To prove our results, we also deduce a few equivalent conditions of two kinds of Hardy-type integral inequalities with a homogeneous kernel in the form of applications. We additionally consider operator expressions. Analytic inequalities of this nature and especially the techniques involved have far reaching applications in various areas in which symmetry plays a prominent role, including aspects of physics and engineering.

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(New Fall 2013:) Matthew de Courcy-Ireland and Peter Humphries have compiled an extensive list of questions for the three standard topics, taken from old general exams. PDF and HTML files are available here:

Algebra: PDF TEX

Real Analysis: PDF TEX

Complex Analysis: PDF TEX

## List of real analysis topics

This is a list of real analysis topics by Wikipedia page

## Basic technique

- Supremum and infimum
- Limit of a sequence
- Monotone convergence theorem
- Cauchy sequence
- Limit superior and limit inferior
- How to evaluate the limit of a real-valued function
- Table of limits
- Oscillation (mathematics)
- Infinite series
- Geometric series
- Harmonic series
- Alternating series
- Telescoping series
- Binomial series
- Summation by parts
- Infinite product
- Pointwise convergence
- Integral test for convergence
- Uniform convergence
- Abel's theorem
- Cesàro mean , Cesàro summation
- Interval (mathematics)
- Vanish at infinity
- Asymptotic notation
- Asymptotic analysis
- Extended real number line
- Partition of an interval
- Cantor set and Cantor space
- Sigma-algebra

## Foundations

- Construction of real numbers
- Dedekind cut

## Fundamental theorems

- Rolle's theorem
- Mean value theorem
- Intermediate value theorem
- Euclidean space
- Heine-Borel theorem
- Bolzano-Weierstrass theorem

## Conditions on real functions

- Continuous function
- Uniform continuity
- Semi-continuity
- Modulus of continuity
- Proper convex function
- An infinitely differentiable function that is not analytic
- Lipschitz continuity
- Dirichlet function
- Darboux function
- Weierstrass function
- Contraction mapping
- Analytic function
- Square-integrable , p-integrable function
- Bounded variation
- Cantor function (example)
- Equicontinuous
- Weakly harmonic

## Inequalities

See list of inequalities

- Bernoulli's inequality
- Generalized mean
- Geometric-harmonic mean
- Harmonic mean
- Weighted mean
- Generalised f-mean
- Arithmetic geometric mean
- Cauchy-Schwarz inequality
- Triangle inequality
- Hölder's inequality
- Minkowski inequality
- Jensen's inequality
- Chebyshev's inequality

## Multivariate calculus

See also list of multivariable calculus topics

- Riemann sum
- Riemann integral
- Fundamental Theorem of Calculus
- Riemann-Stieltjes integral
- Improper integral
- Cauchy principal value
- Darboux integral
- Gamma function
- Rectifiable curve
- Path integral
- An elegant rearrangement of a conditionally convergent iterated integral
- Power series
- Lagrange inversion theorem
- Orthogonal polynomials
- Directional derivative
- Banach fixed point theorem
- Inverse function theorem
- Implicit function theorem
- Lagrange multiplier

For topics relating to measure theory and the Lebesgue integral , see list of integration and measure theory topics ; and also list of probability theory topics

## Distributions

- Dirac delta function
- Heaviside step function
- Hilbert transform
- Green's function

## Fractional calculus

- Differintegral

## Fourier series , Fourier transform

See list of harmonic analysis and representation theory topics

See list of fractal topics

- Hausdorff dimension
- Sierpinski triangle
- Cantor dust
- Sierpinski carpet
- Menger sponge
- Michel Rolle ( 1652 - 1719 )
- Joseph Louis Lagrange ( 1736 - 1813 )
- Jean Baptiste Joseph Fourier ( 1768 - 1830 )
- Augustin Cauchy ( 1789 - 1857 )
- Johann Peter Gustav Lejeune Dirichlet ( 1805 - 1859 )
- Karl Weierstrass ( 1815 - 1897 )
- Bernhard Riemann ( 1826 - 1866 )
- Richard Dedekind ( 1831 - 1916 )
- Rudolf Lipschitz ( 1832 - 1903 )
- Georg Cantor
- Pafnuty Chebyshev
- Camille Jordan
- Hermann Minkowski ( 1864 - 1909 )
- Henri Lebesgue
- Emile Borel
- Felix Hausdorff
- Waclaw Sierpinski
- Karl Menger
- Johann Radon

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## Baftas 2024: the complete list of winners

Every prize at the British Academy Film awards from the Royal Festival Hall in London

- Fox, Grant and Perry: who were the real stars of this year’s Baftas?
- Oppenheimer takes top Baftas
- Peter Bradshaw’s verdict
- The Bafta ceremony and backstage – in pictures

## Baftas 2024: the red carpet, the ceremony, the winners – as it happened

Anatomy of a Fall The Holdovers Killers of the Flower Moon Oppenheimer – WINNER! Poor Things

## Outstanding British film

All of Us Strangers How to Have Sex Napoleon The Old Oak Poor Things Rye Lane Saltburn Scrapper Wonka The Zone of Interest – WINNER!

## Outstanding debut by a British writer, director or producer

Blue Bag Life – Lisa Selby (director), Rebecca Lloyd-Evans (director, producer), Alex Fry (producer) Bobi Wine: The People’s President – Christopher Sharp (director) [also directed by Moses Bwayo] Earth Mama – Savanah Leaf (writer, director, producer), Shirley O’Connor (producer), Medb Riordan (producer) – WINNER! How to Have Sex – Molly Manning Walker (writer, director) Is There Anybody Out There? – Ella Glendining (director)

## Best film not in the English language

20 Days in Mariupol Anatomy of a Fall Past Lives Society of the Snow The Zone of Interest – WINNER!

## Best documentary

20 Days in Mariupol – WINNER! American Symphony Beyond Utopia Still: A Michael J Fox Movie Wham!

## Best animated film

The Boy and the Heron – WINNER! Chicken Run: Dawn of the Nugget Elemental Spider-Man: Across the Spider-Verse

## Best director

Andrew Haigh, All of Us Strangers Justine Triet, Anatomy of a Fall Alexander Payne, The Holdovers Bradley Cooper, Maestro Christopher Nolan, Oppenheimer – WINNER! Jonathan Glazer, The Zone of Interest

## Best original screenplay

Anatomy of a Fall – WINNER! Barbie The Holdovers Maestro Past Lives

## Best adapted screenplay

All of Us Strangers American Fiction – WINNER! Oppenheimer Poor Things The Zone of Interest

## Best leading actress

Fantasia Barrino, The Color Purple Sandra Hüller, Anatomy of a Fall Carey Mulligan, Maestro Vivian Oparah, Rye Lane Margot Robbie, Barbie Emma Stone, Poor Things – WINNER!

## Best leading actor

Bradley Cooper, Maestro Colman Domingo, Rustin Paul Giamatti, The Holdovers Barry Keoghan, Saltburn Cillian Murphy , Oppenheimer – WINNER! Teo Yoo, Past Lives

## Best supporting actress

Emily Blunt, Oppenheimer Danielle Brooks, The Color Purple Claire Foy, All of Us Strangers Sandra Hüller, The Zone of Interest Rosamund Pike, Saltburn Da’ Vine Joy Randolph , The Holdovers – WINNER!

## Best supporting actor

Robert De Niro, Killers of the Flower Moon Robert Downey Jr, Oppenheimer – WINNER! Jacob Elordi, Saltburn Ryan Gosling, Barbie Paul Mescal, All of Us Strangers Dominic Sessa, The Holdovers

## Best casting

All of Us Strangers Anatomy of a Fall The Holdovers – WINNER! How to Have Sex Killers of the Flower Moon

## Best cinematography

Killers of the Flower Moon Maestro Oppenheimer – WINNER! Poor Things The Zone of Interest

## Best editing

Anatomy of a Fall Killers of the Flower Moon Oppenheimer – WINNER! Poor Things The Zone of Interest

## Best costume design

Barbie Killers of the Flower Moon Napoleon Oppenheimer Poor Things – WINNER!

## Best makeup and hair

Killers of the Flower Moon Maestro Napoleon Oppenheimer Poor Things – WINNER!

## Best original score

Killers of the Flower Moon Oppenheimer – WINNER! Poor Things Saltburn Spider-Man: Across the Spider-Verse

## Best production design

Barbie Killers of the Flower Moon Oppenheimer Poor Things – WINNER! The Zone of Interest

Ferrari Maestro Mission: Impossible – Dead Reckoning Part One Oppenheimer The Zone of Interest – WINNER!

## Best special visual effects

The Creator Guardians of the Galaxy Vol 3 Mission: Impossible – Dead Reckoning Part One Napoleon Poor Things – WINNER!

## Best British short animation

Crab Day – WINNER! Visible Mending Wild Summon

## Best British short film

Festival of Slaps Gorka Jellyfish and Lobster – WINNER! Such a Lovely Day Yellow

## EE Rising Star award (voted for by the public)

Phoebe Dynevor Ayo Edebiri Jacob Elordi Mia Mc Kenna-Bruce – WINNER! Sophie Wilde

- Baftas 2024
- Awards and prizes

## Emma Stone embraces method dressing at Baftas with Bella Baxter-inspired look

## Backstage at the 2024 Baftas – in pictures

## Baftas 2024: Emma Stone, Gillian Anderson and Cillian Murphy on the big night – in pictures

## Hugh Grant, Michael J Fox – and Matthew Perry: who were the real stars of this year’s Baftas?

## ‘A must-watch show’: David Tennant to present this year’s film Baftas

## Peter Bradshaw’s Baftas predictions: which films will win – and which should?

## Christopher Nolan set for triumphant Baftas homecoming with Oppenheimer

Most viewed.

Read our research on: Immigration & Migration | Podcasts | Election 2024

## Regions & Countries

How americans view the conflicts between russia and ukraine, israel and hamas, and china and taiwan.

Two years on from Russia’s invasion of Ukraine , 74% of Americans view the war there as important to U.S. national interests – with 43% describing it as very important.

Similar shares see the war between Israel and Hamas (75%) and tensions between China and Taiwan (75%) as important to U.S. national interests, according to a Pew Research Center survey conducted Jan. 22-28.

Pew Research Center conducted this analysis to understand Americans’ views of three ongoing global conflicts: the war between Russia and Ukraine, the war between Israel and Hamas and tensions between China and Taiwan. We first asked respondents to rate how important each conflict is to them personally. We then asked them to rate how important each conflict is to U.S. national interests.

For this analysis, we surveyed 5,146 U.S. adults from Jan. 22 to 28, 2024. Everyone who took part in this survey is a member of the Center’s American Trends Panel (ATP), an online survey panel that is recruited through national, random sampling of residential addresses. This way, nearly all U.S. adults have a chance of selection. The survey is weighted to be representative of the U.S. adult population by gender, race, ethnicity, partisan affiliation, education and other categories. Read more about the ATP’s methodology.

Here are the questions used for this analysis , along with responses, and its methodology .

When asked how important each conflict is to them personally , 59% of Americans say the war between Russia and Ukraine is important to them.

This is similar to the share who say tensions between China and Taiwan (57%) are important to them personally. But it is lower than the share who see the Israel-Hamas war as personally important (65%).

Roughly a third of Americans describe the Israel-Hamas war as very important to them personally, compared with around a quarter for the other two conflicts we asked about.

## Differences by party

Democrats and Democratic-leaning independents are more likely than Republicans and Republican leaners to see the Russia-Ukraine war as important to U.S. national interests (81% vs. 69%).

Related: About half of Republicans now say the U.S. is providing too much aid to Ukraine

However, Democrats and Republicans are about equally likely to see the Israel-Hamas war (76% vs. 77%) and China-Taiwan tensions (76% vs. 78%) as important to U.S. interests.

Americans at the ideological poles – that is, conservative Republicans and liberal Democrats – are more likely than their more moderate counterparts in each party to view both the Israel-Hamas war and China-Taiwan tensions as important to U.S. interests.

When it comes to the importance of each conflict to them personally , Democrats are more likely than Republicans to say the Russia-Ukraine war is important to them (65% vs. 56%), while Republicans are more likely than Democrats to say this about China-Taiwan tensions (62% vs. 56%). Roughly equal shares of Democrats (67%) and Republicans (66%) say the Israel-Hamas war is personally important to them.

Related: Americans’ Views of the Israel-Hamas War

## Differences by age

For all three conflicts we asked about, the oldest Americans are more likely than younger Americans to perceive them as important to both U.S. national interests and to them personally.

However, even among U.S. adults under 30, a majority (58%) see the Israel-Hamas war as personally important. This is not the case for the Russia-Ukraine war or for the ongoing tensions between China and Taiwan.

Note: Here are the questions used for this analysis , along with responses, and its methodology .

## Sign up for our weekly newsletter

Fresh data delivered Saturday mornings

## Majority of U.S. Public Says Trump’s Approach on Iran Has Raised Chances of a Major Conflict

Majorities of u.s. veterans, public say the wars in iraq and afghanistan were not worth fighting, a new perspective on americans’ views of israelis and palestinians, americans divided over decision to withdraw from syria, how people in india see pakistan, 70 years after partition, most popular.

About Pew Research Center Pew Research Center is a nonpartisan fact tank that informs the public about the issues, attitudes and trends shaping the world. It conducts public opinion polling, demographic research, media content analysis and other empirical social science research. Pew Research Center does not take policy positions. It is a subsidiary of The Pew Charitable Trusts .

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## List of real analysis topics

This is a list of articles that are considered real analysis topics.

- 1.2.1 Summation methods
- 1.2.2 More advanced topics
- 1.3.1 Convergence tests
- 1.4.1 Continuity
- 1.4.2 Distributions
- 1.4.3 Variation
- 1.5.1 Differentiation rules
- 1.5.2 Differentiation in geometry and topology
- 1.6.1 Integration and measure theory
- 2 Fundamental theorems
- 3.1.1 Real numbers
- 3.1.2 Specific numbers
- 4.1 Infinite expressions
- 4.2 Inequalities
- 4.4 Orthogonal polynomials
- 4.6 Measures
- 4.7 Field of sets
- 5 Historical figures
- 6 Related fields of analysis

## General topics

- Subsequential limit – the limit of some subsequence
- One-sided limit – either of the two limits of functions of real variables x, as x approaches a point from above or below
- Squeeze theorem – confirms the limit of a function via comparison with two other functions
- Big O notation – used to describe the limiting behavior of a function when the argument tends towards a particular value or infinity, usually in terms of simpler functions

## Sequences and series

( see also list of mathematical series )

- Generalized arithmetic progression – a sequence of numbers such that the difference between consecutive terms can be one of several possible constants
- Geometric progression – a sequence of numbers such that each consecutive term is found by multiplying the previous one by a fixed non-zero number
- Harmonic progression – a sequence formed by taking the reciprocals of the terms of an arithmetic progression
- Finite sequence – see sequence
- Infinite sequence – see sequence
- Divergent sequence – see limit of a sequence or divergent series
- Cauchy sequence – a sequence whose elements become arbitrarily close to each other as the sequence progresses
- Convergent series – a series whose sequence of partial sums converges
- Divergent series – a series whose sequence of partial sums diverges

- Binomial series – the Maclaurin series of the function f given by f ( x ) = (1 + x ) α
- Telescoping series
- Alternating series
- Divergent geometric series
- Harmonic series
- Fourier series
- Lambert series

## Summation methods

- Cesàro summation
- Euler summation
- Lambert summation
- Borel summation
- Summation by parts – transforms the summation of products of into other summations
- Cesàro mean
- Abel's summation formula

## More advanced topics

- Cauchy product –is the discrete convolution of two sequences
- Farey sequence – the sequence of completely reduced fractions between 0 and 1
- Oscillation – is the behaviour of a sequence of real numbers or a real-valued function, which does not converge, but also does not diverge to +∞ or −∞; and is also a quantitative measure for that.
- Indeterminate forms – algebraic expressions gained in the context of limits. The indeterminate forms include 0 0 , 0/0, 1 ∞ , ∞ − ∞, ∞/∞, 0 × ∞, and ∞ 0 .

## Convergence

- Pointwise convergence , Uniform convergence
- Absolute convergence , Conditional convergence
- Normal convergence
- Radius of convergence

## Convergence tests

- Integral test for convergence
- Cauchy's convergence test
- Direct comparison test
- Limit comparison test
- Alternating series test
- Dirichlet's test
- Stolz–Cesàro theorem – is a criterion for proving the convergence of a sequence
- Function of a real variable
- Real multivariable function
- Nowhere continuous function
- Weierstrass function
- Quasi-analytic function
- Non-analytic smooth function
- Flat function
- Bump function
- Differentiable function
- Square-integrable function , p-integrable function
- Bernstein's theorem on monotone functions – states that any real-valued function on the half-line [0, ∞) that is totally monotone is a mixture of exponential functions
- Inverse function
- Convex function , Concave function
- Singular function
- Weakly harmonic function
- Proper convex function
- Rational function
- Orthogonal function
- Implicit function theorem – allows relations to be converted to functions
- Measurable function
- Baire one star function
- Symmetric function
- Differential of a function
- Modulus of continuity
- Lipschitz continuity
- Semi-continuity
- Equicontinuous
- Absolute continuity
- Hölder condition – condition for Hölder continuity

## Distributions

- Dirac delta function
- Heaviside step function
- Hilbert transform
- Green's function
- Bounded variation
- Total variation

## Derivatives

- Inflection point – found using second derivatives
- Directional derivative , Total derivative , Partial derivative

## Differentiation rules

- Linearity of differentiation
- Product rule
- Quotient rule
- Inverse function theorem – gives sufficient conditions for a function to be invertible in a neighborhood of a point in its domain, also gives a formula for the derivative of the inverse function

## Differentiation in geometry and topology

see also List of differential geometry topics

- Differentiable manifold
- Differentiable structure
- Submersion – a differentiable map between differentiable manifolds whose differential is everywhere surjective

(see also Lists of integrals )

- Fundamental theorem of calculus – a theorem of antiderivatives
- Multiple integral
- Iterated integral
- Cauchy principal value – method for assigning values to certain improper integrals
- Line integral
- Anderson's theorem – says that the integral of an integrable, symmetric, unimodal, non-negative function over an n -dimensional convex body ( K ) does not decrease if K is translated inwards towards the origin

## Integration and measure theory

see also List of integration and measure theory topics

- Riemann–Stieltjes integral
- Darboux integral
- Lebesgue integration

## Fundamental theorems

- Monotone convergence theorem – relates monotonicity with convergence
- Intermediate value theorem – states that for each value between the least upper bound and greatest lower bound of the image of a continuous function there is at least one point in its domain that the function maps to that value
- Rolle's theorem – essentially states that a differentiable function which attains equal values at two distinct points must have a point somewhere between them where the first derivative is zero
- Mean value theorem – that given an arc of a differentiable curve, there is at least one point on that arc at which the derivative of the curve is equal to the "average" derivative of the arc

- L'Hôpital's rule – uses derivatives to help evaluate limits involving indeterminate forms
- Abel's theorem – relates the limit of a power series to the sum of its coefficients
- Lagrange inversion theorem – gives the Taylor series of the inverse of an analytic function
- Darboux's theorem – states that all functions that result from the differentiation of other functions have the intermediate value property: the image of an interval is also an interval
- Heine–Borel theorem – sometimes used as the defining property of compactness

## Foundational topics

Real numbers.

- Natural number
- Rational number
- Irrational number
- Completeness of the real numbers
- Least-upper-bound property
- Extended real number line
- Dedekind cut

## Specific numbers

- Neighbourhood
- Derived set (mathematics)
- Completeness
- Partition of an interval
- Contraction mapping
- Fixed point – a point of a function that maps to itself

## Applied mathematical tools

Infinite expressions.

- Continued fraction
- Infinite products

## Inequalities

See list of inequalities

- Triangle inequality
- Bernoulli's inequality
- Cauchy–Schwarz inequality
- Hölder's inequality
- Minkowski inequality
- Jensen's inequality
- Chebyshev's inequality
- Inequality of arithmetic and geometric means
- Generalized mean
- Arithmetic mean
- Geometric mean
- Harmonic mean
- Geometric–harmonic mean
- Arithmetic–geometric mean
- Weighted mean
- Quasi-arithmetic mean

## Orthogonal polynomials

- Hermite polynomials
- Laguerre polynomials
- Jacobi polynomials
- Gegenbauer polynomials
- Legendre polynomials
- Euclidean space
- Banach fixed point theorem – guarantees the existence and uniqueness of fixed points of certain self-maps of metric spaces, provides method to find them
- Complete metric space
- Sequence space
- Compact space
- Lebesgue measure
- Hausdorff measure
- Dominated convergence theorem – provides sufficient conditions under which two limit processes commute, namely Lebesgue integration and almost everywhere convergence of a sequence of functions.

## Field of sets

- Sigma-algebra

## Historical figures

- Michel Rolle (1652–1719)
- Brook Taylor (1685–1731)
- Leonhard Euler (1707–1783)
- Joseph-Louis Lagrange (1736–1813)
- Joseph Fourier (1768–1830)
- Bernard Bolzano (1781–1848)
- Augustin Cauchy (1789–1857)
- Niels Henrik Abel (1802–1829)
- Peter Gustav Lejeune Dirichlet (1805–1859)
- Karl Weierstrass (1815–1897)
- Eduard Heine (1821–1881)
- Pafnuty Chebyshev (1821–1894)
- Leopold Kronecker (1823–1891)
- Bernhard Riemann (1826–1866)
- Richard Dedekind (1831–1916)
- Rudolf Lipschitz (1832–1903)
- Camille Jordan (1838–1922)
- Jean Gaston Darboux (1842–1917)
- Georg Cantor (1845–1918)
- Ernesto Cesàro (1859–1906)
- Otto Hölder (1859–1937)
- Hermann Minkowski (1864–1909)
- Alfred Tauber (1866–1942)
- Felix Hausdorff (1868–1942)
- Émile Borel (1871–1956)
- Henri Lebesgue (1875–1941)
- Wacław Sierpiński (1882–1969)
- Johann Radon (1887–1956)
- Karl Menger (1902–1985)

## Related fields of analysis

- Asymptotic analysis – studies a method of describing limiting behaviour
- List of convexity topics
- List of harmonic analysis topics
- List of Fourier analysis topics
- List of Fourier-related transforms
- Complex analysis – studies the extension of real analysis to include complex numbers
- Functional analysis – studies vector spaces endowed with limit-related structures and the linear operators acting upon these spaces
- Nonstandard analysis – studies mathematical analysis using a rigorous treatment of infinitesimals .
- Calculus , the classical calculus of Newton and Leibniz .
- Non-standard calculus , a rigorous application of infinitesimals , in the sense of non-standard analysis , to the classical calculus of Newton and Leibniz.
- Articles with short description
- Short description with empty Wikidata description
- Real analysis
- Outlines of mathematics and logic
- Wikipedia outlines
- Mathematics-related lists

## IMAGES

## VIDEO

## COMMENTS

Convergence Pointwise convergence, Uniform convergence Absolute convergence, Conditional convergence Normal convergence Radius of convergence Integral test for convergence Cauchy's convergence test Ratio test Direct comparison test Limit comparison test Root test Alternating series test Dirichlet's test

CreateSpace Independent Publishing Platform, 2018. ISBN: 9781718862401. [JL] = Basic Analysis: Introduction to Real Analysis (Vol. 1) (PDF - 2.2MB) by Jiří Lebl, June 2021 (used with permission) This book is available as a free PDF download. You can purchase a paper copy by following a link at the same site.

Chapter 1 The Real Numbers 1 1.1 The Real Number System 1 1.2 Mathematical Induction 10 1.3 The Real Line 19 Chapter 2 Diﬀerential Calculus of Functions of One Variable 30 2.1 Functions and Limits 30 2.2 Continuity 53 2.3 Diﬀerentiable Functions of One Variable 73 2.4 L'Hospital's Rule 88 2.5 Taylor's Theorem 98 Chapter 3 Integral Calculus of Fu...

The real numbers. In real analysis we need to deal with possibly wild functions on R and fairly general subsets of R, and as a result a rm ground-ing in basic set theory is helpful. We begin with the de nition of the real numbers. There are at least 4 di erent reasonable approaches. The axiomatic approach. As advocated by Hilbert, the real ...

Real analysis is an area of analysis that studies concepts such as sequences and their limits, continuity, differentiation, integration and sequences of functions. By definition, real analysis focuses on the real numbers, often including positive and negative infinity to form the extended real line.

Course Description. This course covers the fundamentals of mathematical analysis: convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity, and the interchange of limit operations. It shows the utility of abstract concepts through a study of real numbers, and ….

List of real analysis topics Real analysis A Absolute continuity Almost periodic function Alternating series Analyst's traveling salesman theorem Approximate limit B Baire function Baire one star function Binomial series Bounded function Bounded variation C Càdlàg Cantor's first set theory article Cantor's intersection theorem Carleman's inequality

This is a list of articles that are considered real analysis topics. Introduction List of real analysis topics General topics Limits Sequences and series Summation methods More advanced topics Convergence Convergence tests Functions Continuity Distributions Variation Derivatives Differentiation rules Differentiation in geometry and topology ...

real analysis and offers a view toward more advanced topics. The last chapter is a selection of special topics that lead to other areas of study. These topics form the germs of excellent projects. The appendices serve as references. Appendix A lists the definitions and theorems of elementary analysis, Appendix B offers a very brief

- Real Analysis Introduction [ edit | edit source] The subject of real analysis is concerned with studying the behavior and properties of functions, sequences, and sets on the real number line, which we denote as the mathematically familiar .

This is a short introduction to the fundamentals of real analysis. Although the prerequisites are few, I have written the text assuming the reader has the level of mathematical maturity of one who has completed the standard sequence of calculus courses, has had some exposure to the ideas of mathematical proof (including induction), and has an acquaintance with such basic ideas as equivalence ...

Real Analysis: Foundations is a ... Dedekind's construction of real numbers, historical notes, and selected topics in algebra. Real Analysis: Foundations is ideal for students at the upper-undergraduate or beginning graduate level who are interested in the logical underpinnings of real analysis. With over 130 exercises, it is suitable for a one ...

Rule #1: Don't psyche yourself out. I think it is fairly common for people to regard analysis as a subject where success is completely based off of mathematical intuition, and you either "have ...

List of topics - Basic Real Analysis Course Preliminary Material. Topology Basics: Real numbers, set topology, metric spaces. Topology and continuous functions Baire category theorem Urysohn theorem, extension theorem Spaces of continuous functions, Stone-Weierstrass theorem, Arzela-Ascoli theorem Measures and Measuring

Derivatives Integrals Fundamental theorems Foundational topics Numbers Sets Maps Applied mathematical tools Infinite expressions Inequalities Means Orthogonal polynomials Spaces Measures Field of sets Historical figures Related fields of analysis See also General topics Limits Limit of a sequence Subsequential limit - the limit of some subsequence

Convergence, limits, continuity, smoothness, differentiability, and integrability are some of the features of real-valued sequences and functions that real analysis explores. Complex analysis, on the other hand, is concerned with the study of complex numbers and their functions.

This is a list of articles that are considered real analysis topics. General topics Limits *Limit of a sequence **Subsequential limit - the limit of some subsequence *Limit of a function (see List of limits for a list of limits of common functions) **One-sided limit - either of the two limits of functions of real variables x, as x approaches a point from above or below **Squeeze theorem ...

List of real analysis topics This is a list of articles that are considered real analysis topics. General topics Limits Limit of a sequence Subsequential limit - the limit of some subsequence Limit of a function (see List of limits for a list of limits of common functions)

General topics Limits. Limit of a sequence. Subsequential limit - the limit of some subsequence; Limit of a function (see List of limits for a list of limits of common functions). One-sided limit - either of the two limits of functions of real variables x, as x approaches a point from above or below; Squeeze theorem - confirms the limit of a function via comparison with two other functions

Distance Sets. This chapter covers set theory. The topics include set algebra, relations, orderings and mappings, countability and sequences, real numbers, sequences and limits, and set classes including monotone classes, rings, fields, and sigma fields. The final section introduces the basic ideas of real analysis including Euclidean distance ...

In 1999, Slava Rytchkov, Akshay Venkatesh and Andy Booker compiled their own list of questions in the standard topics. Algebra questions. Real analysis questions. Complex analysis questions. There is also an even older list, compiled by Kiran Kedlaya, containing a few of the most common questions in the standard topics.

List of real analysis topics This is a list of real analysis topics by Wikipedia page NB The topics are in a deliberately chosen order, for the use of students. Basic technique Sequence Supremum and infimum Limit of a sequence Monotone convergence theorem Cauchy sequence Limit superior and limit inferior Limit (mathematics)

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For this analysis, we surveyed 5,146 U.S. adults from Jan. 22 to 28, 2024. Everyone who took part in this survey is a member of the Center's American Trends Panel (ATP), an online survey panel that is recruited through national, random sampling of residential addresses.

David Ingles and Yvonne Man bring you the latest news and analysis to get you ready for the trading day. ... About two dozen banks in the US had portfolios of commercial real estate loans in late ...

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General topics Limits. Limit of a sequence. Subsequential limit - the limit of some subsequence; Limit of a function (see List of limits for a list of limits of common functions). One-sided limit - either of the two limits of functions of real variables x, as x approaches a point from above or below; Squeeze theorem - confirms the limit of a function via comparison with two other functions

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