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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 measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
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 calle ...
s and
series Series may refer to: People with the name * Caroline Series (born 1951), English mathematician, daughter of George Series * George Series (1920–1995), English physicist Arts, entertainment, and media Music * Series, the ordered sets used in ...
of real numbers, and
real function In mathematical analysis, and applications in geometry, applied mathematics, engineering, and natural sciences, a function of a real variable is a function whose domain is the real numbers \mathbb, or a subset of \mathbb that contains an interv ...
s. 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 Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
, continuity, smoothness,
differentiability In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...
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 ...
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 measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
system, which must be established. The real number system consists of an
uncountable set In mathematics, an uncountable set (or uncountably infinite set) 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 numb ...
(\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, an internal binary op ...
s denoted and , and an order denoted . The operations make the real numbers a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
, 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. The basic example of an ordered field is the field of real numbers, and every Dedekind-complete ordered fiel ...
. The real number system is the unique ''
complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...
ordered field'', in the sense that any other complete ordered field is
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
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 mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. The basic example of an ordered field is the field of real numbers, and every Dedekind-complete ordered fiel ...
, 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 In mathematics, the least-upper-bound property (sometimes called completeness or supremum property or l.u.b. property) is a fundamental property of the real numbers. More generally, a partially ordered set has the least-upper-bound property if eve ...
:
''Every nonempty subset of \mathbb that has an upper bound has a
least upper bound In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest lo ...
that is also a real number.''
These
order-theoretic Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article intro ...
properties lead to a number of fundamental results in real analysis, such as the
monotone convergence theorem In the mathematical field of real analysis, the monotone convergence theorem is any of a number of related theorems proving the convergence of monotonic sequences (sequences that are non-decreasing or non-increasing) that are also bounded. Inform ...
, 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 (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
and
operator theory In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operators ...
generalize properties of the real numbers – such generalizations include the theories of
Riesz space In mathematics, a Riesz space, lattice-ordered vector space or vector lattice is a partially ordered vector space where the order structure is a lattice. Riesz spaces are named after Frigyes Riesz who first defined them in his 1928 paper ''Su ...
s and
positive operator In mathematics (specifically linear algebra, operator theory, and functional analysis) as well as physics, a linear operator A acting on an inner product space is called positive-semidefinite (or ''non-negative'') if, for every x \in \mathop(A), \l ...
s. Also, mathematicians consider
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
and
imaginary part 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 ...
s 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 geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
, the real numbers has a ''standard topology'', which is the
order topology In mathematics, an order topology is a certain topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets. If ''X'' is a totally ordered set, t ...
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 is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
function as the real numbers become the prototypical example of a
metric space In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
. 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 Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
whose
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined **Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function * Do ...
is a
countable In mathematics, a set is countable if either it is 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 from it into the natural numbers; ...
,
totally ordered In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexive ...
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(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 Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
(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 Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
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 calle ...
"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 mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
(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,
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...
s, and
integral 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 i ...
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 Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of ma ...
, at the end of the 17th century, for building
infinitesimal calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
. 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 la ...
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(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 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 mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in . Intuitively, a space is complete if there are no "points missing" from it (inside or at the boun ...
''. 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 Below may refer to: *Earth *Ground (disambiguation) *Soil *Floor *Bottom (disambiguation) Bottom may refer to: Anatomy and sex * Bottom (BDSM), the partner in a BDSM who takes the passive, receiving, or obedient role, to that of the top or ...
) 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 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 analysis". Despite leaving university without a degree, he studied mathematics ...
is generally credited for clearly defining the concept of uniform convergence and fully investigating its implications.


Compactness

Compactness is a concept from
general topology In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geomet ...
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 whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a cl ...
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 In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883. Thr ...
\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_ Function_or_functionality_may_refer_to: __Computing_ *_Function_key,_a_type_of_key_on_computer_keyboards *_Function_model,_a_structured_representation_of_processes_in_a_system *_Function_object_or_functor_or_functionoid,_a_concept_of_object-oriente_...
_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_distance,_duration_or_temperature._Here,_''continuous''_means_that_values_can_have_arbitrarily_small_variations._Every_real_...
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_ Function_or_functionality_may_refer_to: __Computing_ *_Function_key,_a_type_of_key_on_computer_keyboards *_Function_model,_a_structured_representation_of_processes_in_a_system *_Function_object_or_functor_or_functionoid,_a_concept_of_object-oriente_...
_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_distance,_duration_or_temperature._Here,_''continuous''_means_that_values_can_have_arbitrarily_small_variations._Every_real_...
s_to_the_real_numbers_can_be_represented_by_a_graph_of_a_function">graph_ Graph_may_refer_to: _Mathematics *Graph_(discrete_mathematics),_a_structure_made_of_vertices_and_edges **Graph_theory,_the_study_of_such_graphs_and_their_properties *Graph_(topology),_a_topological_space_resembling_a_graph_in_the_sense_of_discre_...
_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 Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
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 distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s to the real numbers can be represented by a graph of a function">graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...
in the Cartesian coordinate system">Cartesian plane A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
; 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 Equivalence or Equivalent may refer to: Arts and entertainment *Album-equivalent unit, a measurement unit in the music industry * Equivalence class (music) *'' Equivalent VIII'', or ''The Bricks'', a minimalist sculpture by Carl Andre *''Equiva ...
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 a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Other ...
I = (a, b) = \, or a
closed interval In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Other ...
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 ] In mathematics, a point ''x'' is called an isolated point of a subset ''S'' (in a topological space ''X'') if ''x'' is an element of ''S'' and there exists a neighborhood of ''x'' which does not contain any other points of ''S''. This is equival ...
, or equivalently, E where every p\in E is a
limit point In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contai ...
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 geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
s (which includes
metric space In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
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 (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
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 In mathematics, the image of a function is the set of all output values it may produce. More generally, evaluating a given function f at each element of a given subset A of its domain produces a set, called the "image of A under (or through) ...
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 distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
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 In elementary mathematics, a number line is a picture of a graduated straight line (geometry), line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real ...
. 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 mathematics, two sets are said to be disjoint sets if they have no element in common. Equivalently, two disjoint sets are sets whose intersection is the empty set.. For example, and are ''disjoint sets,'' while and are not disjoint. A ...
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 Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
: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 function In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
s 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 that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex an ...
s, and is strictly contained in C^\infty (see
bump function In mathematics, a bump function (also called a test function) is a function f: \R^n \to \R on a Euclidean space \R^n which is both smooth (in the sense of having continuous derivatives of all orders) and compactly supported. The set of all bump f ...
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 calle ...
(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 In mathematics, the Riemann series theorem (also called the Riemann rearrangement theorem), named after 19th-century German mathematician Bernhard Riemann, says that if an infinite series of real numbers is conditionally convergent, then its terms ...
'' for further discussion). An example of a convergent series is a
geometric series In mathematics, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms. For example, the series :\frac \,+\, \frac \,+\, \frac \,+\, \frac \,+\, \cdots is geometric, because each succ ...
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 Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
or
complex-valued function Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...
''ƒ''(''x'') that is
infinitely differentiable In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if it ...
at a
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
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'' is the
power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a const ...
:f(a) + \frac (x-a) + \frac (x-a)^2 + \frac (x-a)^3 + \cdots. which can be written in the more compact
sigma notation In mathematics, summation is the addition of a sequence of any kind of numbers, called ''addends'' or ''summands''; the result is their ''sum'' or ''total''. Beside numbers, other types of values can be summed as well: functions, vectors, matr ...
as : \sum_ ^ \frac \, (x-a)^ where ''n''! denotes the
factorial In mathematics, the factorial of a non-negative denoted is the 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 (n-1) \times (n-2) \t ...
of ''n'' and ''ƒ'' (''n'')(''a'') denotes the ''n''th
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...
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 , 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 In mathematics, the radius of convergence of a power series is the radius of the largest disk at the center of the series in which the series converges. It is either a non-negative real number or \infty. When it is positive, the power series co ...
, and sums to the function in the
disc of convergence In mathematics, the radius of convergence of a power series is the radius of the largest disk at the center of the series in which the series converges. It is either a non-negative real number or \infty. When it is positive, the power series co ...
, 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 exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, a ...
, 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 o ...
, 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 is a function that repeats its values at regular intervals. For example, the trigonometric functions, which repeat at intervals of 2\pi radians, are periodic functions. Periodic functions are used throughout science to desc ...
s or periodic signals into the sum of a (possibly infinite) set of simple oscillating functions, namely
sines and cosines Sines () is a city and a municipality in Portugal. The municipality, divided into two parishes, has around 14,214 inhabitants (2021) in an area of . Sines holds an important oil refinery and several petrochemical industries. It is also a popular ...
(or
complex exponential The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, al ...
s). 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, Fourier analysis () is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Josep ...
.


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 The method of exhaustion (; ) is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape. If the sequence is correctly constructed, the difference in area bet ...
''. 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 sum In mathematics, a Riemann sum is a certain kind of approximation of an integral by a finite sum. It is named after nineteenth century German mathematician Bernhard Riemann. One very common application is approximating the area of functions or lin ...
s of functions with respect to tagged partitions of an interval. Let ,b/math> be a
closed interval In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Other ...
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 The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each time) with the concept of integrating a function (calculating the area under its graph, or ...
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 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 expressing it through a set o ...
.


Distributions

Distributions (or
generalized functions In mathematics, generalized functions are objects extending the notion of functions. There is more than one recognized theory, for example the theory of distributions. Generalized functions are especially useful in making discontinuous functions ...
) are objects that generalize
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
s. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any
locally integrable In mathematics, a locally integrable function (sometimes also called locally summable function) is a function which is integrable (so its integral is finite) on every compact subset of its domain of definition. The importance of such functions li ...
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 Integration may refer to: Biology *Multisensory integration *Path integration * Pre-integration complex, viral genetic material used to insert a viral genome into a host genome *DNA integration, by means of site-specific recombinase technology, ...
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 distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
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 ancient Greeks, the philosophical nature of infinity was the subject of many discussions amo ...
to form the
extended real line In mathematics, the affinely extended real number system is obtained from the real number system \R by adding two infinity elements: +\infty and -\infty, where the infinities are treated as actual numbers. It is useful in describing the algebra ...
. 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 ...
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 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 derivati ...
, which have a number of useful properties, such as repeated differentiability, expressibility as
power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a const ...
, and satisfying the
Cauchy integral formula In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary o ...
. In real analysis, it is usually more natural to consider
differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
,
smooth Smooth may refer to: Mathematics * Smooth function, a function that is infinitely differentiable; used in calculus and topology * Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions * Smooth algebrai ...
, or
harmonic functions In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f: U \to \mathbb R, where is an open subset of that satisfies Laplace's equation, that is, : \f ...
, which are more widely applicable, but may lack some more powerful properties of holomorphic functions. However, results such as the
fundamental theorem of algebra The fundamental theorem of algebra, also known as d'Alembert's theorem, or the d'Alembert–Gauss theorem, states that every non- constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomia ...
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 In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane. Contour integration is closely related to the calculus of residues, a method of complex analysis. ...
.


Important results

Important results include the Bolzano–Weierstrass and Heine–Borel theorems, the intermediate value theorem and mean value theorem,
Taylor's theorem In calculus, Taylor's theorem gives an approximation of a ''k''-times differentiable function around a given point by a polynomial of degree ''k'', called the ''k''th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the t ...
, the
fundamental theorem of calculus The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each time) with the concept of integrating a function (calculating the area under its graph, or ...
, the Arzelà-Ascoli theorem, the Stone-Weierstrass theorem,
Fatou's lemma In mathematics, Fatou's lemma establishes an inequality relating the Lebesgue integral of the limit inferior of a sequence of functions to the limit inferior of integrals of these functions. The lemma is named after Pierre Fatou. Fatou's lemma ...
, and the monotone convergence and
dominated convergence theorem In measure theory, Lebesgue's dominated convergence theorem provides sufficient conditions under which almost everywhere convergence of a sequence of functions implies convergence in the ''L''1 norm. Its power and utility are two of the primary ...
s.


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 together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
s and
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
s connects real analysis to the field of
general topology In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geomet ...
, 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 (pronounced ) 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 vector ...
s and
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
s 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 (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
.
Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( , ;  – January 6, 1918) was a German mathematician. He played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of ...
'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 Naive set theory is any of several theories of sets used in the discussion of the foundations of mathematics. Unlike Set theory#Axiomatic set theory, axiomatic set theories, which are defined using Mathematical_logic#Formal_logical_systems, forma ...
. 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 In mathematics, Fourier analysis () is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Josep ...
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 derivativ ...
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 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 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 i ...
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 mass and probability of events. These seemingly distinct concepts have many simil ...
. Finally, the generalization of integration from the real line to curves and surfaces in higher dimensional space brought about the study of
vector calculus Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subject ...
, whose further generalization and formalization played an important role in the evolution of the concepts of
differential form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
s and smooth (differentiable) manifolds in
differential geometry Differential geometry is a 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 multili ...
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 c ...
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 ...
.


See also

*
List of real analysis topics This is a list of articles that are considered real analysis topics. General topics Limits *Limit of a sequence ** Subsequential limit – the limit of some subsequence *Limit of a function (''see List of limits for a list of limits of common f ...
*
Time-scale calculus In mathematics, time-scale calculus is a unification of the theory of difference equations with that of differential equations, unifying integral and differential calculus with the calculus of finite differences, offering a formalism for studying ...
– a unification of real analysis with calculus of finite differences *
Real multivariable function In mathematical analysis and its applications, a function of several real variables or real multivariate function is a function with more than one argument, with all arguments being real variables. This concept extends the idea of a function of ...
*
Real coordinate space In mathematics, the real coordinate space of dimension , denoted ( ) or is the set of the -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 vector ...
*
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 ...


References


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 and Functional Analysis
by
Gerald Teschl Gerald Teschl (born 12 May 1970 in Graz) is an Austrian mathematical physicist and professor of mathematics. He works in the area of mathematical physics; in particular direct and inverse spectral theory with application to completely integrable ...
, University of Vienna. {{Analysis-footer