real analysis

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real analysis
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Image:Fourier Series.svg|thumb|200px|The first four partial sums of the Fourier series for a square wavesquare waveIn mathematics, real analysis is the branch of mathematical analysis that studies the behavior of real numbers, sequences and series of real numbers, and real-valued functions.WEB,weblink Lecture notes for MATH 131AH, Tao, Terence, 2003, Course Website for MATH 131AH, Department of Mathematics, UCLA, Some particular properties of real-valued sequences and functions that real analysis studies include convergence, limits, continuity, smoothness, differentiability and integrability.Real analysis is distinguished from complex analysis, which deals with the study of complex numbers and their functions.


{{cleanup|reason=This section goes too heavily into detail about each concept. It should just portray a brief overview in relation to the field of real analysis|date=June 2019}}

Construction of the real numbers

The theorems of real analysis rely intimately upon the structure of the real number line. The real number system consists of a set (mathbb{R}), together with two binary operations denoted {{math|+}} and {{math|â‹…}}, and an order denoted {{math|0, there exists delta>0 such that for all xin E, 00, there exists a natural number N such that m,ngeq N implies that |a_m-a_n|0 of f for every value of xin E, whenever ngeq N, for some integer N. For a family of functions to uniformly converge, sometimes denoted f_nrightrightarrows f, such a value of N must exist for any epsilon>0 given, no matter how small. Intuitively, we can visualize this situation by imagining that, for a large enough N, the functions f_N, f_{N+1}, f_{N+2},ldots are all confined within a 'tube' of width 2epsilon about f (i.e., between f-epsilon and f+epsilon) for every value in their domain E. The distinction between pointwise and uniform convergence is important when exchanging the order of two limiting operations (e.g., taking a limit, a derivative, or integral) is desired: in order for the exchange to be well-behaved, many theorems of real analysis call for uniform convergence. 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. (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. In mathbb{R}, sets that are closed and bounded, and therefore compact, include the empty set, any finite number of points, closed intervals, and their finite unions. However, this list is not exhaustive; for instance, the set {1/n:ninmathbb{N}}cup {0} is another example of a compact set. On the other hand, the set {1/n:ninmathbb{N}} is not compact because it is bounded but not closed, as the boundary point 0 is not a member of the set. The set [0,infty) is also not compact because it is closed but not bounded.For subsets of the real numbers, there are several equivalent definitions of compactness.Definition. A set Esubsetmathbb{R} is compact if it is closed and bounded.This definition also holds for Euclidean space of any finite dimension, mathbb{R}^n, but it is not valid for metric spaces in general. 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).Definition. A set E in a metric space is compact if every sequence in E has a convergent subsequence.This particular property is known as subsequential compactness. In mathbb{R}, a set is subsequentially compact if and only if it is closed and bounded, making this definition equivalent to the one given above. Subsequential compactness is equivalent to the definition of compactness based on subcovers for metric spaces, but not for topological spaces in general.The most general definition of compactness relies on the notion of open covers and subcovers, which is applicable to topological spaces (and thus to metric spaces and mathbb{R} as special cases). In brief, a collection of open sets U_{alpha} is said to be an open cover of set X if the union of these sets is a superset of X. This open cover is said to have a finite subcover if a finite subcollection of the U_{alpha} could be found that also covers X.Definition. A set X in a topological space is compact if every open cover of X has a finite subcover.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.


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. 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. In the first definition given below, f:Itomathbb{R} is a function defined on a non-degenerate interval I of the set of real numbers as its domain. Some possibilities include I=mathbb{R}, the whole set of real numbers, an open interval I = (a, b) = {x in mathbb R ,|, a < x < b }, or a closed interval I = [a, b] = {x in mathbb R ,|, a leq x leq b}. Here, a and b are distinct real numbers, and we exclude the case of I being empty or consisting of only one point, in particular.Definition. If Isubset mathbb{R} is a non-degenerate interval, we say that f:Itomathbb{R} is continuous at pin E if lim_{xto p} f(x) = f(p). We say that f is a continuous map if f is continuous at every pin I.In contrast to the requirements for f to have a limit at a point p, which do not constrain the behavior of f at p itself, the following two conditions, in addition to the existence of lim_{xto p} f(x), must also hold in order for f to be continuous at p: (i) f must be defined at p, i.e., p is in the domain of f; and (ii) f(x)to f(p) as xto p. The definition above actually applies to any domain E that does not contain an isolated point, or equivalently, E where every pin E is a limit point of E. A more general definition applying to f:Xtomathbb{R} with a general domain Xsubset mathbb{R} is the following:Definition. If X is an arbitrary subset of mathbb{R}, we say that f:Xtomathbb{R} is continuous at pin X if, for any epsilon>0, there exists delta>0 such that for all xin X, |x-p|0 such that for all x,yin X, |x-y|0, for a given epsilon>0.

Absolute continuity

Definition. Let Isubsetmathbb{R} be an interval on the real line. A function f:I to mathbb{R} is said to be absolutely continuous on I if for every positive number epsilon, there is a positive number delta such that whenever a finite sequence of pairwise disjoint sub-intervals (x_1, y_1), (x_2,y_2),ldots, (x_n,y_n) of I satisfies{{harvnb|Royden|1988|loc=Sect. 5.4, page 108}}; {{harvnb|Nielsen|1997|loc=Definition 15.6 on page 251}}; {{harvnb|Athreya|Lahiri|2006|loc=Definitions 4.4.1, 4.4.2 on pages 128,129}}. The interval I is assumed to be bounded and closed in the former two books but not the latter book.
sum_{k=1}^{n} (y_k - x_k) < delta
displaystyle sum_{k=1}^{n} | f(y_k) - f(x_k) | < epsilon.
Absolutely continuous functions are continuous: consider the case n = 1 in this definition. 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. This approximation, if it exists, is unique and is given by the line that is tangent to the function at the given point a, and the slope of the line is the derivative of the function at a. A function f:mathbb{R}tomathbb{R} is differentiable at a if the limit
f'(a)=lim_{hto 0}frac{f(a+h)-f(a)}{h}
exists. This limit is known as the derivative of f at a, and the function f', possibly defined on only a subset of mathbb{R}, is the derivative (or derivative function) of f. If the derivative exists everywhere, the function is said to be differentiable.As a simple consequence of the definition, f is continuous at a if it is differentiable there. Differentiability is therefore a stronger regularity condition (condition describing the "smoothness" of a function) than continuity, and it is possible for a function to be continuous on the entire real line but not differentiable anywhere (see Weierstrass's nowhere differentiable continuous function). It is possible to discuss the existence of higher-order derivatives as well, by finding the derivative of a derivative function, and so on.One can classify functions by their differentiability class. The class C^0 (sometimes C^0([a,b]) to indicate the interval of applicability) consists of all continuous functions. The class C^1 consists of all differentiable functions whose derivative is continuous; such functions are called continuously differentiable. Thus, a C^1 function is exactly a function whose derivative exists and is of class C^0. In general, the classes C^k can be defined recursively by declaring C^0 to be the set of all continuous functions and declaring C^k for any positive integer k to be the set of all differentiable functions whose derivative is in C^{k-1}. In particular, C^k is contained in C^{k-1} for every k, and there are examples to show that this containment is strict. Class C^infty is the intersection of the sets C^k as k varies over the non-negative integers, and the members of this class are known as the smooth functions. Class C^omega consists of all analytic functions, and is strictly contained in C^infty (see bump function for a smooth function that is not analytic).The chain rule, mean value theorem, l'Hospital's rule, and Taylor's theorem are important results in the elementary theory of the derivative.


A series formalizes the imprecise notion of taking the sum of an endless sequence of numbers. The idea that taking the sum of an "infinite" number of terms can lead to a finite result was counterintuitive to the ancient Greeks and led to the formulation of a number of paradoxes by Zeno and other philosophers. The modern notion of assigning a value to a series avoids dealing with the ill-defined notion of adding an "infinite" number of terms. Instead, the finite sum of the first n terms of the sequence, known as a partial sum, is considered, and the concept of a limit is applied to the sequence of partial sums as n grows without bound. The series is assigned the value of this limit, if it exists. Given an (infinite) sequence (a_n), we can define an associated series as the formal mathematical object a_1+a_2+a_3+cdots=sum_{n=1}^{infty} a_n, sometimes simply written as sum a_n. The partial sums of a series sum a_n are the numbers s_n=sum_{j=1}^n a_j. A series sum a_n is said to be convergent if the sequence consisting of its partial sums, (s_n), is convergent; otherwise it is divergent. The sum of a convergent series is defined as the number s=lim_{ntoinfty}s_n.It is to be emphasized that the word "sum" is used here in a metaphorical sense as a shorthand for taking the limit of a sequence of partial sums and should not be interpreted as simply "adding" an infinite number of terms. For instance, in contrast to the behavior of finite sums, rearranging the terms of an infinite series may result in convergence to a different number (see the article on the Riemann rearrangement theorem for further discussion).An example of a convergent series is a geometric series which forms the basis of one of Zeno's famous paradoxes:
sum_{n=1}^infty frac{1}{2^n} = frac{1}{2}+ frac{1}{4}+ frac{1}{8}+cdots=1.
In contrast, the harmonic series has been known since the Middle Ages to be a divergent series:
(Here, "=infty" is merely a notational convention to indicate that the partial sums of the series grow without bound.)A series sum a_n is said to converge absolutely if sum |a_n| is convergent. A convergent series sum a_n for which sum |a_n| diverges is said to converge conditionally (or nonabsolutely). It is easily shown that absolute convergence of a series implies its convergence. On the other hand, an example of a conditionally convergent series is
sum_{n=1}^inftyfrac{(-1)^{n-1}}{n}=1-frac{1}{2}+frac{1}{3}-frac{1}{4}+cdots=log 2.

Taylor series

The Taylor series of a real or complex-valued function Æ’(x) that is infinitely differentiable at a real or complex number a is the power series
f(a)+frac {f'(a)}{1!} (x-a)+ frac{f''(a)}{2!} (x-a)^2+frac{f^{(3)}(a)}{3!}(x-a)^3+ cdots.
which can be written in the more compact sigma notation as
sum_{n=0} ^ {infty} frac {f^{(n)}(a)}{n!} , (x-a)^{n}
where n! denotes the factorial of n and Æ’ (n)(a) denotes the nth derivative of Æ’ evaluated at the point a. The derivative of order zero Æ’ is defined to be Æ’ itself and {{nowrap|(x − a)0}} and 0! are both defined to be 1. In the case that {{nowrap|a {{=}} 0}}, the series is also called a Maclaurin series.A Taylor series of f about point a may diverge, converge at only the point a, converge for all x such that |x-a| mathematical analysis > Fourier analysis.


Integration is a formalization of the problem of finding the area bound by a curve and the related problems of determining the length of a curve or volume enclosed by a surface. The basic strategy to solving problems of this type was known to the ancient Greeks and Chinese, and was known as the method of exhaustion. Generally speaking, the desired area is bounded from above and below, respectively, by increasingly accurate circumscribing and inscribing polygonal approximations whose exact areas can be computed. By considering approximations consisting of a larger and larger ("infinite") number of smaller and smaller ("infinitesimal") pieces, the area bound by the curve can be deduced, as the upper and lower bounds defined by the approximations converge around a common value. The spirit of this basic strategy can easily be seen in the definition of the Riemann integral, in which the integral is said to exist if upper and lower Riemann (or Darboux) sums converge to a common value as thinner and thinner rectangular slices ("refinements") are considered. Though the machinery used to define it is much more elaborate compared to the Riemann integral, the Lebesgue integral was defined with similar basic ideas in mind. Compared to the Riemann integral, the more sophisticated Lebesgue integral allows area (or length, volume, etc.; termed a "measure" in general) to be defined and computed for much more complicated and irregular subsets of Euclidean space, although there still exist "non-measurable" subsets for which an area cannot be assigned.

Riemann integration

The Riemann integral is defined in terms of Riemann sums of functions with respect to tagged partitions of an interval. Let [a,b] be a closed interval of the real line; then a tagged partition cal{P} of [a,b] is a finite sequence
a = x_0 le t_1 le x_1 le t_2 le x_2 le cdots le x_{n-1} le t_n le x_n = b . ,!
This partitions the interval [a,b] into n sub-intervals [x_{i-1},x_i] indexed by i=1,ldots, n, each of which is "tagged" with a distinguished point t_iin[x_{i-1},x_i]. For a function f bounded on [a,b], we define the Riemann sum of f with respect to tagged partition cal{P} as
sum_{i=1}^{n} f(t_i) Delta_i,
where Delta_i=x_i-x_{i-1} is the width of sub-interval i. Thus, each term of the sum is the area of a rectangle with height equal to the function value at the distinguished point of the given sub-interval, and width the same as the sub-interval width. The mesh of such a tagged partition is the width of the largest sub-interval formed by the partition, ||Delta_i||=max_{i=1,ldots, n}Delta_i. We say that the Riemann integral of f on [a,b] is S if for any epsilon>0 there exists delta>0 such that, for any tagged partition cal{P} with mesh ||Delta_i||

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