real analysis

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OR:

In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the branch of real analysis studies the behavior of
real number In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
s,
sequence In mathematics, a sequence is an enumerated collection of mathematical object, objects in which repetitions are allowed and order theory, order matters. Like a Set (mathematics), set, it contains Element (mathematics), members (also called ''eleme ...
s 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. 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 Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
, 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 form a ...
s and their functions.

# Scope

## Construction of the real numbers

The theorems of real analysis rely on the properties of the
real number In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
system, which must be established. The real number system consists of an uncountable set ($\mathbb$), together with two binary operations denoted and , and an order denoted . The operations make the real numbers a field, and, along with the order, an ordered field. 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' 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 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 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, 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 (mathematics), sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative number, negative (in which cas ...
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), functio ...
. 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 those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...
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 $a\left(n\right) = a_n$ 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 $\left(a_n\right)$ is ''bounded'' if there exists $M\in\R$ such that

## 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 mathematical object, objects in which repetitions are allowed and order theory, order matters. Like a Set (mathematics), set, it contains Element (mathematics), members (also called ''eleme ...
"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, originally called infinitesimal calculus or "the calculus of infinitesimals", 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 generalizati ...
(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 (m ...
in general) and its formal definition is used in turn to define notions like continuity, derivatives, and
integral In mathematics, an integral assigns numbers to Function (mathematics), functions in a way that describes Displacement (geometry), displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding ...
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 Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He w ...
, and made rigorous, at the end of the 19th century by
Bolzano Bolzano ( or ; german: Bozen, (formerly ); bar, Bozn; lld, Balsan or ) is the capital city of the province of South Tyrol in northern Italy. With a population of 108,245, Bolzano is also by far the largest city in South Tyrol and the third ...
and
Weierstrass Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern mathematical analysis, analysis". Despite leaving university without a degree, ...
, who gave the modern ε-δ definition, which follows. Definition. Let $f$ be a real-valued function defined on We say that ''$f\left(x\right)$ tends to $L$ as $x$ approaches $x_0$'', or that ''the limit of $f\left(x\right)$ 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 $, f\left(x\right) - L, < \varepsilon$. 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 $f\left(x\right)\to L$ 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 $f\left(x\right)$ 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 $f\left(x_0\right)$ 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 $\left(a_n\right)$ when $n$ becomes large. Definition. Let $\left(a_n\right)$ be a real-valued sequence. We say that $\left(a_n\right)$ ''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 $\left(a_n\right)$ fails to converge, we say that $\left(a_n\right)$ ''diverges''. Generalizing to a real-valued function of a real variable, a slight modification of this definition (replacement of sequence $\left(a_n\right)$ and term $a_n$ by function $f$ and value $f\left(x\right)$ and natural numbers $N$ and $n$ by real numbers $M$ and $x$, respectively) yields the definition of the ''limit of $f\left(x\right)$ 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 $f\left(x\right)$ 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 $\left(a_n\right)$ be a real-valued sequence. We say that $\left(a_n\right)$ 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 it is convergent. This property of the real numbers is expressed by saying that the real numbers endowed with the standard metric, $\left(\R, , \cdot, \right)$, 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 $\left(f_n\right)_^\infty$, 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 $x\in E$, $f_n\left(x\right)\to f\left(x\right)$ 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 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

## _Continuity

A__function_from_the_set_of_real_number_ In_mathematics,_a_real_number_is_a_number_that_can_be_used_to_measurement,_measure_a_''continuous''_one-dimensional_quantity_such_as_a_distance,_time,_duration_or_temperature._Here,_''continuous''_means_that_values_can_have_arbitrarily_small_var_...
s_to_the_real_numbers_can_be_represented_by_a_ 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_measurement,_measure_a_''continuous''_one-dimensional_quantity_such_as_a_distance,_time,_duration_or_temperature._Here,_''continuous''_means_that_values_can_have_arbitrarily_small_var_...
s_to_the_real_numbers_can_be_represented_by_a_graph_of_a_function">graph_in_the_Cartesian_coordinate_system.html" ;"title="graph_of_a_function.html" ;"title="Heine–Borel theorem">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 measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
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_ graph_in_the_Cartesian_coordinate_system">Cartesian_plane;_such_a_function_is_continuous_if,_roughly_speaking,_the_graph_is_a_single_unbroken_curve">graph_of_a_function">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_Equivalence_relation.html" ;"title="curve.html" ;"title="graph of a function">graph in the Cartesian coordinate system">Cartesian plane; such a function is continuous if, roughly speaking, the graph is a single unbroken curve">graph of a function">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 Equivalence relation">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 $I = \left(a, b\right) = \,$ or a closed interval $I = \left[a, b\right] = \.$ 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) $f\left(x\right)\to f\left(p\right)$ 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 $, f\left(x\right)-f\left(p\right), < \varepsilon$. 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 spaces (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), functio ...
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 $f^ \left(V\right)$ is a
neighborhood A neighbourhood (British English British English (BrE, en-GB, or BE) is, according to Oxford Dictionaries, " English as used in Great Britain, as distinct from that used elsewhere". More narrowly, it can refer specifically to the En ...
of $p$ in $X$ for every neighborhood $V$ of $f\left(p\right)$ in $Y$. We say that $f$ is a ''continuous map'' if $f^\left(U\right)$ is open in $X$ for every $U$ open in $Y$. (Here, $f^\left(S\right)$ 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 measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
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 $, f\left(x\right)-f\left(y\right), < \varepsilon$. 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 $f:\left(0,1\right)\to\mathbb$ defined by $f\left(x\right)=1/x$. By choosing points close to 0, we can always make $, f\left(x\right)-f\left(y\right), > \varepsilon$ 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 sub-intervals $\left(x_1, y_1\right), \left(x_2,y_2\right),\ldots, \left(x_n,y_n\right)$ of $I$ satisfies :$\sum_^ \left(y_k - x_k\right) < \delta$ then :$\sum_^ , f\left(y_k\right) - f\left(x_k\right) , < \varepsilon.$ 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 :$f\text{'}\left(a\right)=\lim_\frac$ exists. This limit is known as the ''derivative of $f$ at $a$'', and the function $f\text{'}$, 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 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 Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematics ...
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 function In mathematics, an analytic function is a function (mathematics), function that is locally given by a convergent series, convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are ...
s, and is strictly contained in $C^\infty$ (see bump function 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 mathematical object, objects in which repetitions are allowed and order theory, order matters. Like a Set (mathematics), set, it contains Element (mathematics), members (also called ''eleme ...
$\left(a_n\right)$, 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, $\left(s_n\right)$, 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 A paradox is a logically self-contradictory statement or a statement that runs contrary to one's expectation. It is a statement that, despite apparently valid reasoning from true premises, leads to a seemingly self-contradictory or a logically u ...
: :$\sum_^\infty \frac = \frac + \frac + \frac + \cdots = 1 .$ 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 ''ƒ''(''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 form a ...
''a'' is the power series :$f\left(a\right) + \frac \left(x-a\right) + \frac \left(x-a\right)^2 + \frac \left(x-a\right)^3 + \cdots.$ which can be written in the more compact sigma notation as :$\sum_ ^ \frac \, \left(x-a\right)^$ 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 ''ƒ'' (''n'')(''a'') denotes the ''n''th derivative of ''ƒ'' evaluated at the point ''a''. The derivative of order zero ''ƒ'' is defined to be ''ƒ'' 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
exponential function The exponential function is a mathematical Function (mathematics), function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponentiation, exponent). Unless otherwise specified, the term generally refers to the positiv ...
, the
logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 of ...
, the trigonometric functions and their inverses are extended to functions of a complex variable.

### Fourier series

Fourier series decomposes periodic functions 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 an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
>
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 (m ...
>
Fourier analysis In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented ...
.

## 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

### 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.

## 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 (plural, : analyses) is the process of breaking a complexity, complex topic or Substance theory, substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics a ...
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 measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
s, often including positive and negative
infinity Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol . Since the time of the Greek mathematics, ancient Greeks, the Infinity (philosophy), philosophical nature of infinit ...
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 Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
, 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 form a ...
s. In complex analysis, it is natural to define differentiation via
holomorphic functions In mathematics, a holomorphic function is a complex-valued function of one or Function of several complex variables, more complex number, complex variables that is Differentiable function#Differentiability in complex analysis, complex differen ...
, 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), functio ...
s and topological spaces 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 spaces and Hilbert spaces and, more generally to functional analysis. Georg Cantor'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 for sequences of functions eventually gave rise to
Fourier analysis In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented ...
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 functions 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 Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
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 spaces, a fundamental concept in measure theory. 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 differential calculus, integral calculus, linear algebra ...
and other closely related areas of
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...
and
topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
.

* 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 of dimension , denoted ( ) or is the set of the tuple, -tuples of real numbers, that is the set of all sequences of real numbers. With component-wise addition and scalar multiplication, it is a real v ...
*
Complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...

# Bibliography

* * * * * * * * * * *

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 and Functional Analysis
by Gerald Teschl, University of Vienna. {{Analysis-footer