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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the branch of real analysis studies the behavior of
real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s,
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...
s and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include
convergence Convergence may refer to: Arts and media Literature *''Convergence'' (book series), edited by Ruth Nanda Anshen *Convergence (comics), "Convergence" (comics), two separate story lines published by DC Comics: **A four-part crossover storyline that ...
, limits, continuity, smoothness, differentiability and integrability. Real analysis is distinguished from
complex analysis 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 ...
, which deals with the study of
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s and their functions.


Scope


Construction of the real numbers

The theorems of real analysis rely on the properties of the (established)
real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
system. The real number system consists of an
uncountable set In mathematics, an uncountable set, informally, is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger t ...
(\mathbb), together with two
binary operation In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, a binary operation ...
s denoted and \cdot, and a total order denoted . The operations make the real numbers a field, and, along with the order, an
ordered field In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. Basic examples of ordered fields are the rational numbers and the real numbers, both with their standard ord ...
. The real number system is the unique '' complete ordered field'', in the sense that any other complete ordered field is isomorphic to it. Intuitively, completeness means that there are no 'gaps' (or 'holes') in the real numbers. This property distinguishes the real numbers from other ordered fields (e.g., the rational numbers \mathbb) and is critical to the proof of several key properties of functions of the real numbers. The completeness of the reals is often conveniently expressed as the ''least upper bound property'' (see below).


Order properties of the real numbers

The real numbers have various lattice-theoretic properties that are absent in the complex numbers. Also, the real numbers form an
ordered field In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. Basic examples of ordered fields are the rational numbers and the real numbers, both with their standard ord ...
, 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:
''Every nonempty subset of \mathbb that has an upper bound has a least upper bound that is also a real number.''
These order-theoretic properties lead to a number of fundamental 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. In particular, many ideas in
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics ...
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.


Topological properties of the real numbers

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. As a
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
, the real numbers has a ''standard topology'', which is the order topology induced by order <. Alternatively, by defining the ''metric'' or ''distance function'' d:\mathbb\times\mathbb\to\mathbb_ using the
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if x is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...
function as the real numbers become the prototypical example of a
metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
. The topology induced by metric d turns out to be identical to the standard topology induced by order <. Theorems like the intermediate value theorem that are essentially topological in nature can often be proved in the more general setting of metric or topological spaces rather than in \mathbb only. Often, such proofs tend to be shorter or simpler compared to classical proofs that apply direct methods.


Sequences

A ''sequence'' is a function whose domain is a
countable In mathematics, a Set (mathematics), set is countable if either it is finite set, finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function fro ...
, totally ordered set. The domain is usually taken to be the
natural number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
s, although it is occasionally convenient to also consider bidirectional sequences indexed by the set of all integers, including negative indices. Of interest in real analysis, a ''real-valued sequence'', here indexed by the natural numbers, is a map a : \N \to \R : n \mapsto a_n. Each a(n) = a_n is referred to as a ''term'' (or, less commonly, an ''element'') of the sequence. A sequence is rarely denoted explicitly as a function; instead, by convention, it is almost always notated as if it were an ordered ∞-tuple, with individual terms or a general term enclosed in parentheses: (a_n) = (a_n)_=(a_1, a_2, a_3, \dots) . A sequence that tends to a limit (i.e., \lim_ a_n exists) is said to be convergent; otherwise it is divergent. (''See the section on limits and convergence for details.'') A real-valued sequence (a_n) is ''bounded'' if there exists M\in\R such that , a_n, for all n\in\mathbb. A real-valued sequence (a_n) is ''monotonically increasing'' or ''decreasing'' if a_1 \leq a_2 \leq a_3 \leq \cdots or a_1 \geq a_2 \geq a_3 \geq \cdots holds, respectively. If either holds, the sequence is said to be ''monotonic''. The monotonicity is ''strict'' if the chained inequalities still hold with \leq or \geq replaced by < or >. Given a sequence (a_n), another sequence (b_k) is a ''subsequence'' of (a_n) if b_k=a_ for all positive integers k and (n_k) is a strictly increasing sequence of natural numbers.


Limits and convergence

Roughly speaking, a limit is the value that a function or a
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...
"approaches" as the input or index approaches some value. (This value can include the symbols \pm\infty when addressing the behavior of a function or sequence as the variable increases or decreases without bound.) The idea of a limit is fundamental to
calculus Calculus is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations. Originally called infinitesimal calculus or "the ...
(and
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series ( ...
in general) and its formal definition is used in turn to define notions like continuity,
derivative In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
s, and
integral In mathematics, an integral is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental oper ...
s. (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 the 17th century, for building infinitesimal calculus. For sequences, the concept was introduced by Cauchy, and made rigorous, at the end of the 19th century by Bolzano and Weierstrass, who gave the modern ε-δ definition, which follows. Definition. Let f be a real-valued function defined on We say that ''f(x) tends to L as x approaches x_0'', or that ''the limit of f(x) as x approaches x_0 is L'' if, for any \varepsilon>0, there exists \delta>0 such that for all x\in E, 0 < , x - x_0, < \delta implies that , f(x) - L, < \varepsilon. We write this symbolically as f(x)\to L\ \ \text\ \ x\to x_0 , or as \lim_ f(x) = L . Intuitively, this definition can be thought of in the following way: We say that f(x)\to L as x\to x_0, when, given any positive number \varepsilon, no matter how small, we can always find a \delta, such that we can guarantee that f(x) and L are less than \varepsilon apart, as long as x (in the domain of f) is a real number that is less than \delta away from x_0 but distinct from x_0. The purpose of the last stipulation, which corresponds to the condition 0<, x-x_0, in the definition, is to ensure that \lim_ f(x)=L does not imply anything about the value of f(x_0) itself. Actually, x_0 does not even need to be in the domain of f in order for \lim_ f(x) to exist. In a slightly different but related context, the concept of a limit applies to the behavior of a sequence (a_n) when n becomes large. Definition. Let (a_n) be a real-valued sequence. We say that (a_n) ''converges to'' a if, for any \varepsilon > 0, there exists a natural number N such that n\geq N implies that , a-a_n, < \varepsilon. We write this symbolically as a_n \to a\ \ \text\ \ n \to \infty ,or as\lim_ a_n = a ; if (a_n) fails to converge, we say that (a_n) ''diverges''. Generalizing to a real-valued function of a real variable, a slight modification of this definition (replacement of sequence (a_n) and term a_n by function f and value f(x) and natural numbers N and n by real numbers M and x, respectively) yields the definition of the ''limit of f(x) as x increases without bound'', notated \lim_ f(x). Reversing the inequality x\geq M to x \leq M gives the corresponding definition of the limit of f(x) as x ''decreases'' ''without bound'', 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. Definition. Let (a_n) be a real-valued sequence. We say that (a_n) is a ''Cauchy sequence'' if, for any \varepsilon > 0, there exists a natural number N such that m,n\geq N implies that , a_m-a_n, < \varepsilon. It can be shown that a real-valued sequence is Cauchy
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
it is convergent. This property of the real numbers is expressed by saying that the real numbers endowed with the standard metric, (\R, , \cdot, ), is a '' complete metric space''. 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.


Uniform and pointwise convergence for sequences of functions

In addition to sequences of numbers, one may also speak of ''sequences of functions'' ''on'' E\subset \mathbb, that is, infinite, ordered families of functions f_n:E\to\mathbb, denoted (f_n)_^\infty, and their convergence properties. 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. Roughly speaking, pointwise convergence of functions f_n to a limiting function f:E\to\mathbb, denoted f_n \rightarrow f, simply means that given any x\in E, f_n(x)\to f(x) as n\to\infty. 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. Uniform convergence requires members of the family of functions, f_n, to fall within some error \varepsilon > 0 of f for ''every value of x\in E'', whenever n\geq N, for some integer N. For a family of functions to uniformly converge, sometimes denoted f_n\rightrightarrows f, such a value of N must exist for any \varepsilon>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_, f_,\ldots are all confined within a 'tube' of width 2\varepsilon about f (that is, between f - \varepsilon and f+\varepsilon) ''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

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 In geometry, topology, and related branches of mathematics, a closed set is a Set (mathematics), set whose complement (set theory), complement is an open set. In a topological space, a closed set can be defined as a set which contains all its lim ...
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, 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 \\cup \\ is a compact set; the Cantor ternary set \mathcal\subset ,1/math> is another example of a compact set. On the other hand, the set \ is not compact because it is bounded but not closed, as the boundary point 0 is not a member of the set. The set 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, 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 as special cases). In brief, a collection of open sets U_ 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_ 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.


Continuity

A function from the set of
real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s to the real numbers can be represented by a graph in the Cartesian coordinate system">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:I\to\R is a function defined on a non-degenerate interval I of the set of real numbers as its domain. Some possibilities include I=\R, the whole set of real numbers, an
open interval In mathematics, a real interval is the set (mathematics), set of all real numbers lying between two fixed endpoints with no "gaps". Each endpoint is either a real number or positive or negative infinity, indicating the interval extends without ...
I = (a, b) = \, or a closed interval I = , 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 I\subset \mathbb is a non-degenerate interval, we say that f:I \to \R is ''continuous at'' p\in I if \lim_ f(x) = f(p). We say that f is a ''continuous map'' if f is continuous at every p\in 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_ 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 x\to p. The definition above actually applies to any domain E that does not contain an isolated point, or equivalently, E where every p\in E is a limit point of E. A more general definition applying to f:X\to\mathbb with a general domain X\subset \mathbb is the following: Definition. If X is an arbitrary subset of \mathbb, we say that f:X\to\mathbb is ''continuous at'' p\in X if, for any \varepsilon>0, there exists \delta>0 such that for all x\in X, , x-p, <\delta implies that , f(x)-f(p), < \varepsilon. We say that f is a ''continuous map'' if f is continuous at every p\in X. A consequence of this definition is that f is ''trivially continuous at any isolated point'' p\in X. This somewhat unintuitive treatment of isolated points is necessary to ensure that our definition of continuity for functions on the real line is consistent with the most general definition of continuity for maps between
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
s (which includes
metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
s and \mathbb in particular as special cases). This definition, which extends beyond the scope of our discussion of real analysis, is given below for completeness. Definition. If X and Y are topological spaces, we say that f:X\to Y is ''continuous at'' p\in X if f^ (V) is a
neighborhood A neighbourhood (Commonwealth English) or neighborhood (American English) is a geographically localized community within a larger town, city, suburb or rural area, sometimes consisting of a single street and the buildings lining it. Neigh ...
of p in X for every neighborhood V of f(p) in Y. We say that f is a ''continuous map'' if f^(U) is open in X for every U open in Y. (Here, f^(S) refers to the preimage of S\subset Y under f.)


Uniform continuity

Definition. If X is a subset of the
real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s, we say a function f:X\to\mathbb is ''uniformly continuous'' ''on'' X if, for any \varepsilon > 0, there exists a \delta>0 such that for all x,y\in X, , x-y, <\delta implies that , f(x)-f(y), < \varepsilon. Explicitly, when a function is uniformly continuous on X, the choice of \delta needed to fulfill the definition must work for ''all of'' X for a given \varepsilon. In contrast, when a function is continuous at every point p\in X (or said to be continuous on X), the choice of \delta may depend on both \varepsilon ''and'' p. In contrast to simple continuity, uniform continuity is a property of a function that only makes sense with a specified domain; to speak of uniform continuity at a single point p is meaningless. On a compact set, it is easily shown that all continuous functions are uniformly continuous. If E is a bounded noncompact subset of \mathbb, then there exists f:E\to\mathbb that is continuous but not uniformly continuous. As a simple example, consider f:(0,1)\to\mathbb defined by f(x)=1/x. By choosing points close to 0, we can always make , f(x)-f(y), > \varepsilon for any single choice of \delta>0, for a given \varepsilon > 0.


Absolute continuity

Definition. Let I\subset\mathbb be an interval on the real line. A function f:I \to \mathbb is said to be ''absolutely continuous'' ''on'' I if for every positive number \varepsilon, there is a positive number \delta such that whenever a finite sequence of
pairwise disjoint In set theory in mathematics and Logic#Formal logic, formal logic, two Set (mathematics), sets are said to be disjoint sets if they have no element (mathematics), element in common. Equivalently, two disjoint sets are sets whose intersection (se ...
sub-intervals (x_1, y_1), (x_2,y_2),\ldots, (x_n,y_n) of I satisfies :\sum_^ (y_k - x_k) < \delta then :\sum_^ , f(y_k) - f(x_k) , < \varepsilon. 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 a fundamental 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.


Differentiation

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\to\mathbb is ''differentiable at a'' if the limit :f'(a)=\lim_\frac exists. This limit is known as the ''derivative of f at a'', and the function f', possibly defined on only a subset of \mathbb, 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( ,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^. In particular, ''C^k'' is contained in C^ 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 In mathematical analysis, a bump function (also called a test function) is a function f : \Reals^n \to \Reals on a Euclidean space \Reals^n which is both smooth (in the sense of having continuous derivatives of all orders) and compactly supp ...
for a smooth function that is not analytic).


Series

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 In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...
(a_n), we can define an associated ''series'' as the formal mathematical object sometimes simply written as \sum a_n. The ''partial sums'' of a series \sum a_n are the numbers s_n=\sum_^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 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_^\infty \frac = \frac + \frac + \frac + \cdots = 1 . In contrast, the harmonic series has been known since the Middle Ages to be a divergent series: :\sum_^\infty \frac = 1 + \frac + \frac + \cdots = \infty . (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'' ''non-absolutely''.The term ''unconditional convergence'' refers to series whose sum does not depend on the order of the terms (i.e., any rearrangement gives the same sum). Convergence is termed ''conditional'' otherwise. For series in \R^n, it can be shown that absolute convergence and unconditional convergence are equivalent. Hence, the term "conditional convergence" is often used to mean non-absolute convergence. However, in the general setting of Banach spaces, the terms do not coincide, and there are unconditionally convergent series that do not converge absolutely. It is easily shown that absolute convergence of a series implies its convergence. On the other hand, an example of a series that converges non-absolutely is :\sum_^\infty \frac = 1 - \frac + \frac - \frac + \cdots = \ln 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 In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
''a'' is the power series :f(a) + \frac (x-a) + \frac (x-a)^2 + \frac (x-a)^3 + \cdots. which can be written in the more compact sigma notation as : \sum_ ^ \frac \, (x-a)^ where ''n''! denotes the
factorial In mathematics, the factorial of a non-negative denoted is the Product (mathematics), product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times ...
of ''n'' and ''Æ’'' (''n'')(''a'') denotes the ''n''th
derivative In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
of ''Æ’'' evaluated at the point ''a''. The derivative of order zero ''Æ’'' is defined to be ''Æ’'' itself and and 0! are both defined to be 1. In the case that , 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, (the largest such ''R'' for which convergence is guaranteed is called the ''radius of convergence''), or converge on the entire real line. Even a converging Taylor series may converge to a value different from the value of the function at that point. If the Taylor series at a point has a nonzero radius of convergence, and sums to the function in the disc of convergence, then the function is analytic. The analytic functions have many fundamental properties. In particular, an analytic function of a real variable extends naturally to a function of a complex variable. It is in this way that the exponential function, the
logarithm In mathematics, the logarithm of a number is the exponent by which another fixed value, the base, must be raised to produce that number. For example, the logarithm of to base is , because is to the rd power: . More generally, if , the ...
, the
trigonometric functions In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all ...
and their inverses are extended to functions of a complex variable.


Fourier series

Fourier series decomposes
periodic function A periodic function, also called a periodic waveform (or simply periodic wave), is a function that repeats its values at regular intervals or periods. The repeatable part of the function or waveform is called a ''cycle''. For example, the t ...
s or periodic signals into the sum of a (possibly infinite) set of simple oscillating functions, namely sines and cosines (or complex exponentials). The study of Fourier series typically occurs and is handled within the branch
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
>
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series ( ...
> Fourier analysis.


Integration

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 ,b/math> be a closed interval of the real line; then a ''tagged partition'' \cal of ,b/math> is a finite sequence : a = x_0 \le t_1 \le x_1 \le t_2 \le x_2 \le \cdots \le x_ \le t_n \le x_n = b . \,\! This partitions the interval ,b/math> into n sub-intervals _,x_i/math> indexed by i=1,\ldots, n, each of which is "tagged" with a distinguished point t_i\in _,x_i/math>. For a function f bounded on ,b/math>, we define the ''Riemann sum'' of f with respect to tagged partition \cal as :\sum_^ f(t_i) \Delta_i, where \Delta_i=x_i-x_ 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_\Delta_i. We say that the ''Riemann integral'' of f on ,b/math> is S if for any \varepsilon>0 there exists \delta>0 such that, for any tagged partition \cal with mesh \, \Delta_i \, < \delta, we have ::\left, S - \sum_^ f(t_i)\Delta_i \ < \varepsilon. This is sometimes denoted \mathcal\int_^b f=S. When the chosen tags give the maximum (respectively, minimum) value of each interval, the Riemann sum is known as the upper (respectively, lower) ''Darboux sum''. A function is ''Darboux integrable'' if the upper and lower Darboux sums can be made to be arbitrarily close to each other for a sufficiently small mesh. Although this definition gives the Darboux integral the appearance of being a special case of the Riemann integral, they are, in fact, equivalent, in the sense that a function is Darboux integrable if and only if it is Riemann integrable, and the values of the integrals are equal. In fact, calculus and real analysis textbooks often conflate the two, introducing the definition of the Darboux integral as that of the Riemann integral, due to the slightly easier to apply definition of the former. The fundamental theorem of calculus asserts that integration and differentiation are inverse operations in a certain sense.


Lebesgue integration and measure

Lebesgue integration is a mathematical construction that extends the integral to a larger class of functions; it also extends the domains on which these functions can be defined. The concept of a ''measure'', an abstraction of length, area, or volume, is central to Lebesgue integral
probability theory Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
.


Distributions

Distributions (or generalized functions) are objects that generalize functions. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative.


Relation to complex analysis

Real analysis is an area of
analysis Analysis (: analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (38 ...
that studies concepts such as sequences and their limits, continuity, differentiation, integration and sequences of functions. By definition, real analysis focuses on the
real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s, often including positive and negative
infinity Infinity is something which is boundless, endless, or larger than any natural number. It is denoted by \infty, called the infinity symbol. From the time of the Ancient Greek mathematics, ancient Greeks, the Infinity (philosophy), philosophic ...
to form the extended real line. Real analysis is closely related to
complex analysis 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 ...
, which studies broadly the same properties of
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s. In complex analysis, it is natural to define differentiation via holomorphic functions, which have a number of useful properties, such as repeated differentiability, expressibility as power series, and satisfying the Cauchy integral formula. In real analysis, it is usually more natural to consider differentiable, smooth, or harmonic functions, which are more widely applicable, but may lack some more powerful properties of holomorphic functions. However, results such as the fundamental theorem of algebra are simpler when expressed in terms of complex numbers. Techniques from the theory of analytic functions of a complex variable are often used in real analysis – such as evaluation of real integrals by residue calculus.


Important results

Important results include the Bolzano–Weierstrass and Heine–Borel theorems, the intermediate value theorem and mean value theorem, Taylor's theorem, the fundamental theorem of calculus, the Arzelà-Ascoli theorem, the Stone-Weierstrass theorem, Fatou's lemma, and the monotone convergence and dominated convergence theorems.


Generalizations and related areas of mathematics

Various ideas from real analysis can be generalized from the real line to broader or more abstract contexts. These generalizations link real analysis to other disciplines and subdisciplines. For instance, generalization of ideas like continuous functions and compactness from real analysis to
metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
s and
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
s connects real analysis to the field of general topology, while generalization of finite-dimensional Euclidean spaces to infinite-dimensional analogs led to the concepts of
Banach space In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and ...
s and Hilbert spaces and, more generally to
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics ...
.
Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( ; ;  â€“ 6 January 1918) was a mathematician who played a pivotal role in the creation of set theory, which has become a foundations of mathematics, fundamental theory in mathematics. Cantor establi ...
's investigation of sets and sequence of real numbers, mappings between them, and the foundational issues of real analysis gave birth to naive set theory. The study of issues of
convergence Convergence may refer to: Arts and media Literature *''Convergence'' (book series), edited by Ruth Nanda Anshen *Convergence (comics), "Convergence" (comics), two separate story lines published by DC Comics: **A four-part crossover storyline that ...
for sequences of functions eventually gave rise to Fourier analysis as a subdiscipline of mathematical analysis. Investigation of the consequences of generalizing differentiability from functions of a real variable to ones of a complex variable gave rise to the concept of
holomorphic 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 . The existence of a complex de ...
s and the inception of
complex analysis 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 ...
as another distinct subdiscipline of analysis. On the other hand, the generalization of integration from the Riemann sense to that of Lebesgue led to the formulation of the concept of abstract
measure space A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that ...
s, a fundamental concept in
measure theory In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude (mathematics), magnitude, mass, and probability of events. These seemingl ...
. Finally, the generalization of integration from the real line to curves and surfaces in higher dimensional space brought about the study of vector calculus, whose further generalization and formalization played an important role in the evolution of the concepts of differential forms and smooth (differentiable) manifolds in
differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...
and other closely related areas of
geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
and
topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
.


See also

* List of real analysis topics * Time-scale calculus – a unification of real analysis with calculus of finite differences * Real multivariable function *
Real coordinate space In mathematics, the real coordinate space or real coordinate ''n''-space, of dimension , denoted or , is the set of all ordered -tuples of real numbers, that is the set of all sequences of real numbers, also known as '' coordinate vectors''. ...
*
Complex analysis 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 ...


References


Sources

* * *


Bibliography

* * * * * * * * * * *


External links


How We Got From There to Here: A Story of Real Analysis
by Robert Rogers and Eugene Boman
A First Course in Analysis
by Donald Yau
Analysis WebNotes
by John Lindsay Orr

by Bert G. Wachsmuth

by John O'Connor

by Elias Zakon

by Elias Zakon *

* ttp://www.jirka.org/ra/ Basic Analysis: Introduction to Real Analysisby Jiri Lebl
Topics in Real Analysis
by Gerald Teschl, University of Vienna. {{Analysis-footer