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[[Image:Fourier Series.svg|thumb|200px|The first four partial sums of the [[Fourier series]] for a [[square wave]]. Fourier series are an important tool in real analysis.]]
'''Real analysis''' (traditionally, the '''theory of functions of a real variable''') is a branch of [[mathematical analysis]] dealing with the [[real number]]s and real-valued functions of a real variable. In particular, it deals with the [[Mathematical analysis|analytic]] properties of real [[function (mathematics)|functions]] and [[sequence]]s, including [[Limit of a sequence|convergence]] and [[limit of a function|limit]]s of [[sequence]]s of real numbers, the [[calculus]] of the real numbers, and [[continuous function|continuity]], [[smooth function|smoothness]] and related properties of real-valued functions.

==Scope==
===Construction of the real numbers===
{{Main|Construction of the real numbers}}
There are several ways of defining the [[real number]] system as an [[ordered field]]. The ''synthetic'' approach gives a list of [[axiom]]s for the real numbers as a ''complete ordered [[field (mathematics)|field]]''. Under the usual axioms of [[axiomatic set theory|set theory]], one can show that these axioms are categorical, in the sense that there is a model for the axioms, and any two such models are [[isomorphic]]. Any one of these models must be explicitly constructed, and most of these models are built using the basic properties of the [[rational number]] system as an ordered field. These constructions are described in more detail in the main article.

=== Order properties of the real numbers ===
The real numbers have several important [[lattice theory|lattice-theoretic]] properties that are absent in the complex numbers. Most importantly, the real numbers form an [[ordered field]], in which addition and multiplication preserve positivity. Moreover, the ordering of the real numbers is [[totally ordered|total]], and the real numbers have the [[least upper bound property]]. These [[partially ordered set|order-theoretic]] properties lead to a number of important results in real analysis, such as the [[monotone convergence theorem]], the [[intermediate value theorem]] and the [[mean value theorem]].

However, while the results in real analysis are stated for real numbers, many of these results can be generalized to other mathematical objects. In particular, many ideas in [[functional analysis]] and [[operator theory]] generalize properties of the real numbers – such generalizations include the theories of [[Riesz space]]s and [[positive operator]]s. Also, mathematicians consider [[real part|real]] and [[imaginary part]]s of complex sequences, or by [[strong operator topology|pointwise evaluation]] of [[operator (mathematics)|operator]] sequences.

===Sequences===
{{Main|Sequence (mathematics)}}
A sequence is usually defined as a [[function (mathematics)|function]] whose domain is a [[countable]] [[totally ordered]] set, although in many disciplines the domain is restricted, such as to the [[natural numbers]]. In [[real analysis]] a sequence is a function from a [[subset]] of the [[natural numbers]] to the [[real numbers]].<ref name="Gaughan">{{cite book|title=Introduction to Analysis |last=Gaughan |first=Edward |publisher=AMS (2009)|ISBN=0-8218-4787-2|chapter=1.1 Sequences and Convergence}}</ref> In other words, a sequence is a map ''f''(''n'') : '''N''' → '''R'''. To recover our earlier notation we might identify ''an'' = ''f''(''n'') {{pad|.5em}} for all ''n'' or just write ''an'' : '''N''' → '''R'''.

===Limits===
{{Main|Limit (mathematics)}}
A '''limit''' is the value that a [[function (mathematics)|function]] or [[sequence]] "approaches" as the input or index approaches some value.<ref>{{cite book|last=Stewart|first=James|authorlink=James Stewart (mathematician)|title=Calculus: Early Transcendentals|publisher=[[Brooks/Cole]]|edition=6th|year=2008|isbn =0-495-01166-5}}</ref> Limits are essential to [[calculus]] (and [[mathematical analysis]] in general) and are used to define [[continuous function|continuity]], [[derivative]]s, and [[integral]]s.

The concept of a [[limit of a sequence]] is further generalized to the concept of a limit of a [[net (topology)|topological net]].
===Continuity===
{{Main|Continuous function}}
A [[function (mathematics)|function]] from the set of [[real number]]s to the real numbers can be represented by a [[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. These definitions are [[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 definitions below,
:<math>f\colon I \rightarrow \mathbf R.</math>
is a function defined on a [[subset]] ''I'' of the set '''R''' of real numbers. This subset ''I'' is referred to as the [[domain of a function|domain]] of ''f''. Some possible choices include ''I''='''R''', the whole set of real numbers, an [[open interval]]
:<math>I = (a, b) = \{x \in \mathbf R \,|\, a < x 
or a [[closed interval]]
:<math>I = [a, b] = \{x \in \mathbf R \,|\, a \leq x \leq b \}. </math>
Here, ''a'' and ''b'' are real numbers.

====Uniform continuity====
{{Main|Uniform continuity}}
If ''X'' and ''Y'' are subsets of the [[real number]]s, a function ''f'' : ''X'' → ''Y'' is called '''uniformly continuous''' if for all ''ε'' > 0 there exists a ''δ'' > 0 such that for all ''x'', ''y'' ∈ ''X'', |''x'' − ''y''| < ''δ'' implies |''f''(''x'') − ''f''(''y'')| < ''ε.

The difference between being uniformly continuous, and being simply continuous at every point, is that in uniform continuity the value of ''δ'' depends only on ''ε'' and not on the point in the domain.

====Absolute continuity====
{{Main|Absolute continuity}}
Let <math>I</math> be an [[interval (mathematics)|interval]] in the [[real line]] '''R'''. A function <math>f:I \to R</math> is '''absolutely continuous''' on <math>I</math> if for every positive number <math>\epsilon</math>, there is a positive number <math>\delta</math> such that whenever a finite sequence of [[pairwise disjoint]] sub-intervals <math>(x_k, y_k)</math> of <math>I</math> satisfies<ref>{{harvnb|Royden|1988|loc=Sect. 5.4, page 108}}; {{harvnb|Nielsen|1997|loc=Definition 15.6 on page 251}}; {{harvnb|Athreya|Lahiri|2006|loc=Definitions 4.4.1, 4.4.2 on pages 128,129}}. The interval ''I'' is assumed to be bounded and closed in the former two books but not the latter book.</ref>
:<math>\sum_{k} \left| y_k - x_k \right| < \delta</math>
then
:<math>\displaystyle \sum_{k} | f(y_k) - f(x_k) | < \epsilon.</math>
The collection of all absolutely continuous functions on ''I'' is denoted AC(''I'').

The following conditions on a real-valued function ''f'' on a compact interval [''a'',''b''] are equivalent:<ref>{{harvnb|Nielsen|1997|loc=Theorem 20.8 on page 354}}; also {{harvnb|Royden|1988|loc=Sect. 5.4, page 110}} and {{harvnb|Athreya|Lahiri|2006|loc=Theorems 4.4.1, 4.4.2 on pages 129,130}}.</ref>

:(1) ''f'' is absolutely continuous;

:(2) ''f'' has a derivative ''f'' ′ [[almost everywhere]], the derivative is Lebesgue integrable, and
:: <math> f(x) = f(a) + \int_a^x f'(t) \, dt </math>
:for all ''x'' on [''a'',''b''];

:(3) there exists a Lebesgue integrable function ''g'' on [''a'',''b''] such that
:: <math> f(x) = f(a) + \int_a^x g(t) \, dt </math>
:for all ''x'' on [''a'',''b''].

If these equivalent conditions are satisfied then necessarily ''g'' = ''f'' ′ almost everywhere.

Equivalence between (1) and (3) is known as the '''fundamental theorem of Lebesgue integral calculus''', due to [[Lebesgue]].<ref>{{harvnb|Athreya|Lahiri|2006|loc=before Theorem 4.4.1 on page 129}}.</ref>
===Series===
{{Main|series (mathematics)}}
Given an [[infinite set|infinite]] [[sequence]] of numbers { ''a''''n'' }, a '''series''' is informally the result of adding all those terms together: ''a''1 + ''a''2 + ''a''3 + · · ·. These can be written more compactly using the [[summation]] symbol ∑. An example is the famous series from [[Zeno's paradoxes#Proposed solutions|Zeno's dichotomy]] and [[1/2 + 1/4 + 1/8 + 1/16 + · · ·|its mathematical representation]]:

:<math>\sum_{n=1}^\infty \frac{1}{2^n} = \frac{1}{2}+ \frac{1}{4}+ \frac{1}{8}+\cdots.</math>

The terms of the series are often produced according to a certain rule, such as by a [[formula]], or by an [[algorithm]].
====Taylor series====
{{Main|Taylor series}}
The Taylor series of a [[real-valued function|real]] or [[complex-valued function]] ''ƒ''(''x'') that is [[infinitely differentiable function|infinitely differentiable]] at a [[real number|real]] or [[complex number]] ''a'' is the [[power series]]
:<math>f(a)+\frac {f'(a)}{1!} (x-a)+ \frac{f''(a)}{2!} (x-a)^2+\frac{f^{(3)}(a)}{3!}(x-a)^3+ \cdots. </math>

which can be written in the more compact [[Summation#Capital-sigma_notation|sigma notation]] as

:<math> \sum_{n=0} ^ {\infty} \frac {f^{(n)}(a)}{n!} \, (x-a)^{n}</math>

where ''n''! denotes the [[factorial]] 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 {{nowrap|(''x'' − ''a'')0}} and 0! are both defined to be 1. In the case that {{nowrap|''a'' {{=}} 0}}, the series is also called a Maclaurin series.

====Fourier Series====
A '''Fourier series''' decomposes [[periodic function]]s or periodic signals into the sum of a (possibly infinite) set of simple oscillating functions, namely [[sine wave|sines and cosines]] (or [[complex exponential]]s). The study of Fourier series is a branch of [[Fourier analysis]].

===Differentiation===
{{Main|Differentiation (mathematics)}}
Formally, the derivative of the function ''f'' at ''a'' is the [[limit of a function|limit]]

:<math>f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}</math>

If the derivative exists everywhere, the function is '''differentiable'''. One can take higher derivatives as well, by iterating this process.

One can classify functions by their '''differentiability class'''. The class ''C''0 consists of all continuous functions. The class ''C''1 consists of all [[differentiable function]]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 ''Ck'' can be defined [[recursion|recursively]] by declaring ''C''0 to be the set of all continuous functions and declaring ''Ck'' for any positive integer ''k'' to be the set of all differentiable functions whose derivative is in ''C''''k''−1. In particular, ''Ck'' is contained in ''C''''k''−1 for every ''k'', and there are examples to show that this containment is strict. ''C''∞ is the intersection of the sets ''Ck'' as ''k'' varies over the non-negative integers. ''C''ω is strictly contained in ''C''∞.

===Integration===
====Riemann integration====
{{Main|Riemann integral}}
The Riemann integral is defined in terms of [[Riemann sum]]s of functions with respect to ''tagged partitions'' of an interval. Let [''a'',''b''] be a [[Interval (mathematics)|closed interval]] of the real line; then a ''tagged partition'' of [''a'',''b''] is a finite sequence

:<math> a = x_0 \le t_1 \le x_1 \le t_2 \le x_2 \le \cdots \le x_{n-1} \le t_n \le x_n = b . \,\!</math>

This partitions the interval [''a'',''b''] into ''n'' sub-intervals {{nowrap|[''x''''i''−1, ''x''''i'']}} indexed by ''i'', each of which is "tagged" with a distinguished point {{nowrap|''t''''i'' ∈ [''x''''i''−1, ''x''''i'']}}. A ''Riemann sum'' of a function ''f'' with respect to such a tagged partition is defined as
:<math>\sum_{i=1}^{n} f(t_i) \Delta_i ; </math>
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. Let {{nowrap|Δ''i'' {{=}} ''x''''i''−''x''''i''−1}} be the width of sub-interval ''i''; then the ''mesh'' of such a tagged partition is the width of the largest sub-interval formed by the partition, {{nowrap|max''i''{{=}}1…''n'' Δ''i''}}. The ''Riemann integral'' of a function ''f'' over the interval [''a'',''b''] is equal to ''S'' if:
:For all {{nowrap|ε > 0}} there exists {{nowrap|δ > 0}} such that, for any tagged partition [''a'',''b''] with mesh less than δ, we have
::<math>\left| S - \sum_{i=1}^{n} f(t_i)\Delta_i \right| < \varepsilon.</math>
When the chosen tags give the maximum (respectively, minimum) value of each interval, the Riemann sum becomes an upper (respectively, lower) [[Darboux integral|Darboux sum]], suggesting the close connection between the Riemann integral and the [[Darboux integral]].

====Lebesgue integration====
{{Main|Lebesgue integral}}
'''Lebesgue integration''' is a mathematical construction that extends the integral to a larger class of functions; it also extends the [[Domain (mathematics)|domain]]s on which these functions can be defined.
===Distributions===
{{Main|Distribution (mathematics)}}
'''Distributions''' (or '''[[generalized functions]]''') are objects that generalize [[function (mathematics)|function]]s. Distributions make it possible to [[derivative|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 [[mathematical analysis|analysis]] that studies concepts such as sequences and their limits, continuity, [[derivative|differentiation]], [[integral|integration]] and sequences of functions. By definition, real analysis focuses on the [[real number]]s, often including positive and negative [[infinity (mathematics)|infinity]] to form the [[extended real line]]. Real analysis is closely related to [[complex analysis]], which studies broadly the same properties of [[complex number]]s. In complex analysis, it is natural to define [[derivative|differentiation]] via [[holomorphic functions]], which have a number of useful properties, such as repeated differentiability, expressability as [[power series]], and satisfying the [[Cauchy integral formula]].

In real analysis, it is usually more natural to consider [[differentiable]], [[smooth functions|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 theorem|residue calculus]].

==Important results==
Important results include the [[Bolzano–Weierstrass theorem|Bolzano–Weierstrass]] and [[Heine–Borel theorem]]s, the [[intermediate value theorem]] and [[mean value theorem]], the [[fundamental theorem of calculus]], and the [[monotone convergence theorem]].

Various ideas from real analysis can be generalized from real space to general [[metric space]]s, as well as to [[measure space]]s, [[Banach space]]s, and [[Hilbert space]]s.

==See also==
* [[List of real analysis topics]]
* [[Time-scale calculus]] – a unification of real analysis with calculus of finite differences
* [[Real multivariable function]]
* [[Real coordinate space]]
* [[Complex analysis]]

==References==
{{reflist}}

==Bibliography==
*{{cite book |last1=Aliprantis |first1=Charalambos D. |authorlink1=Charalambos D. Aliprantis |last2=Burkinshaw |first2=Owen |title=Principles of real analysis |edition=3rd |publisher=Academic |year=1998 |isbn=0-12-050257-7}}
*{{cite book |last=Browder |first=Andrew |title=Mathematical Analysis: An Introduction |series=Undergraduate Texts in Mathematics |location=New York |publisher=Springer-Verlag |year=1996 |isbn=0-387-94614-4}}
*{{cite book |last1=Bartle |first1=Robert G. |authorlink1=Robert G. Bartle |last2=Sherbert |first2=Donald R. |title=Introduction to Real Analysis |edition=3rd |location=New York |publisher=John Wiley and Sons |year=2000 |isbn=0-471-32148-6}}
*{{cite book |last=Abbott |first=Stephen |title=Understanding Analysis |series=Undergradutate Texts in Mathematics |isbn=0-387-95060-5 |year=2001 |location=New York |publisher=Springer-Verlag}}
*{{cite book |last=Rudin |first=Walter |authorlink=Walter Rudin |title=Principles of Mathematical Analysis |series=Walter Rudin Student Series in Advanced Mathematics |edition=3rd |publisher=McGraw–Hill |isbn=978-0-07-054235-8}}
*{{cite book |last1=Dangello |first1=Frank |last2=Seyfried |first2=Michael |title=Introductory Real Analysis |isbn=978-0-395-95933-6 |publisher=Brooks Cole |year=1999}}
*{{cite book |last=Bressoud |first=David |authorlink=David Bressoud |title=A Radical Approach to Real Analysis |isbn=0-88385-747-2 |publisher=MAA |year=2007}}
*{{cite book |last1=Kolmogorov |first1=A. N. |authorlink1=Andrey Kolmogorov |last2=Fomin |first2=S. V. |authorlink2=Sergei Fomin |others=Translated by Richard A. Silverman |title=Introductory Real Analysis |year=1975 | publisher=Dover Publications |url=http://store.doverpublications.com/0486612260.html |accessdate=2 April 2013 |isbn=0486612260}}

==External links==
* [http://www.worldscientific.com/worldscibooks/10.1142/8580 A First Course in Analysis] by Donald Yau
* [http://www.math.unl.edu/~webnotes/contents/chapters.htm Analysis WebNotes] by John Lindsay Orr
* [http://www.mathcs.org/analysis/reals/index.html Interactive Real Analysis] by Bert G. Wachsmuth
* [http://www-groups.mcs.st-andrews.ac.uk/~john/analysis/index.html A First Analysis Course] by John O'Connor
* [http://www.trillia.com/zakon-analysisI.html Mathematical Analysis I] by Elias Zakon
* [http://www.trillia.com/zakon-analysisII.html Mathematical Analysis II] by Elias Zakon
* {{Cite book | last1=Trench | first1=William F. | title=Introduction to Real Analysis | url=http://ramanujan.math.trinity.edu/wtrench/texts/TRENCH_REAL_ANALYSIS.PDF | publisher=[[Prentice Hall]] | isbn=978-0-13-045786-8 | year=2003 | postscript=}}
* [http://www.economics.soton.ac.uk/staff/aldrich/Calculus%20and%20Analysis%20Earliest%20Uses.htm Earliest Known Uses of Some of the Words of Mathematics: Calculus & Analysis]
* [http://www.jirka.org/ra/ Basic Analysis: Introduction to Real Analysis] by Jiri Lebl
* [http://www.mat.univie.ac.at/~gerald/ftp/book-fa/index.html Topics in Real and Functional Analysis] by [[Gerald Teschl]], University of Vienna.

[[Category:Real analysis| ]]

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en>Monkbot: /* References */Fix CS1 deprecated date parameter errors