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Most importantly, the real numbers form an ordered field , in which sums and products of positive numbers are also positive. Moreover, the ordering of the real numbers is total , and the real numbers have the least upper bound property :. These order-theoretic properties lead to a number of important results in real analysis, such as the monotone convergence theorem , the intermediate value theorem and the mean value theorem.

However, while the results in real analysis are stated for real numbers, many of these results can be generalized to other mathematical objects.

Real analysis

In particular, many ideas in functional analysis and operator theory generalize properties of the real numbers — such generalizations include the theories of Riesz spaces and positive operators. Also, mathematicians consider real and imaginary parts of complex sequences, or by pointwise evaluation of operator sequences. Many of the theorems of real analysis are consequences of the topological properties of the real number line. The order properties of the real numbers described above are closely related to these topological properties.

Often, such proofs tend to be shorter or simpler compared to classical proofs that apply direct methods. A sequence is a function whose domain is a countable , totally ordered set. The domain is usually taken to be the natural numbers [2] , although it is occasionally convenient to also consider bidirectional sequences indexed by the set of all integers, including negative indices. A sequence that tends to a limit i. See the section on limits and convergence for details. If either holds, the sequence is said to be monotonic. Roughly speaking, a limit is the value that a function or a sequence "approaches" as the input or index approaches some value.

The idea of a limit is fundamental to calculus and mathematical analysis in general and its formal definition is used in turn to define notions like continuity , derivatives , and integrals. In fact, the study of limiting behavior has been used as a characteristic that distinguishes calculus and mathematical analysis from other branches of mathematics.

The concept of limit was informally introduced for functions by Newton and Leibniz , at the end of 17th century, for building infinitesimal calculus. We write this symbolically as. Sometimes, it is useful to conclude that a sequence converges, even though the value to which it converges is unknown or irrelevant. In these cases, the concept of a Cauchy sequence is useful.

It can be shown that a real-valued sequence is Cauchy if and only if it is convergent. In a general metric space, however, a Cauchy sequence need not converge. In addition, for real-valued sequences that are monotonic, it can be shown that the sequence is bounded if and only if it is convergent. However, in the case of sequences of functions, there are two kinds of convergence, known as pointwise convergence and uniform convergence , that need to be distinguished.

In contrast, uniform convergence is a stronger type of convergence, in the sense that a uniformly convergent sequence of functions also converges pointwise, but not conversely.

Real Analysis: An Introduction to the Theory of Real Functions and Integration

The distinction between pointwise and uniform convergence is important when exchanging the order of two limiting operations e. For example, a sequence of continuous functions see below is guaranteed to converge to a continuous limiting function if the convergence is uniform, while the limiting function may not be continuous if convergence is only pointwise. Karl Weierstrass is generally credited for clearly defining the concept of uniform convergence and fully investigating its implications.

Compactness is a concept from general topology that plays an important role in many of the theorems of real analysis. The property of compactness is a generalization of the notion of a set being closed and bounded.

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In the context of real analysis, these notions are equivalent: a set in Euclidean space is compact if and only if it is closed and bounded. Briefly, a closed set contains all of its boundary points , while a set is bounded if there exists a real number such that the distance between any two points of the set is less than that number.

The equivalence of the definition with the definition of compactness based on subcovers, given later in this section, is known as the Heine-Borel theorem. A more general definition that applies to all metric spaces uses the notion of a subsequence see above. This particular property is known as subsequential compactness.

Subsequential compactness is equivalent to the definition of compactness based on subcovers for metric spaces, but not for topological spaces in general. Compact sets are well-behaved with respect to properties like convergence and continuity. For instance, any Cauchy sequence in a compact metric space is convergent. As another example, the image of a compact metric space under a continuous map is also compact.

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A function from the set of real numbers to the real numbers can be represented by a graph in the Cartesian plane ; such a function is continuous if, roughly speaking, the graph is a single unbroken curve with no "holes" or "jumps". There are several ways to make this intuition mathematically rigorous.

Several definitions of varying levels of generality can be given.


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In cases where two or more definitions are applicable, they are readily shown to be equivalent to one another, so the most convenient definition can be used to determine whether a given function is continuous or not. This definition, which extends beyond the scope of our discussion of real analysis, is given below for completeness. On a compact set, it is easily shown that all continuous functions are uniformly continuous. The collection of all absolutely continuous functions on I is denoted AC I.

Absolute continuity is an important concept in the Lebesgue theory of integration, allowing the formulation of a generalized version of the fundamental theorem of calculus that applies to the Lebesgue integral. The notion of the derivative of a function or differentiability originates from the concept of approximating a function near a given point using the "best" linear approximation.

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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. This text has a lot of essential and useful figures and formulas.

I believe the figures and graphs make students understand more easily.

Real Analysis - Wikibooks, open books for an open world

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.


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  • There are also some drawbacks to the book like ordering the topics. Nevertheless, I value this book in teaching the course Analysis. This is a short introduction to the fundamentals of real analysis. See, in particular, Clark's freely downloadable PDF expositions on commutative Real Analysis Exchange - Michigan State University Press The Real Analysis Exchange journal has four sections: conventional research articles; topical surveys, which give an overview of one area of current research activity; "Inroads," intended for a less traditional presentation of information of interest The Tutor World Online journal posts free primers in math.

    Video Lectures in Mathematics - Jerry Farlow An online pinboard gathering thousands of visual links to a broad range of math videos. Categories include applications of math; famous mathematicians; math education; math humor; math on TV, movies; STEM math initiatives; and "instructional: K Blog posts, which date to February, , have included "Riemannian manifolds and curvature" and "Sailing into the wind, or faster than the wind" Will the real continuous function please stand up?

    All rights reserved. Interactive Real Analysis - Bert G.