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| {{Calculus}}
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| In [[mathematics]], a '''continuous function''' is a [[function (mathematics)|function]] for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be a "'''discontinuous function'''". A continuous function with a continuous [[inverse function]] is called "[[bicontinuous]]".
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| Continuity of functions is one of the core concepts of [[topology]], which is treated in full generality below. The introductory portion of this article focuses on the special case where the inputs and outputs of functions are [[real number]]s. In addition, this article discusses the definition for the more general case of functions between two [[metric space]]s. In [[order theory]], especially in [[domain theory]], one considers a notion of continuity known as [[Scott continuity]]. Other forms of continuity do exist but they are not discussed in this article.
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| As an example, consider the function ''h''(''t''), which describes the [[height]] of a growing flower at time ''t''. This function is continuous. In fact, a dictum of [[classical physics]] states that ''in nature everything is continuous''. By contrast, if ''M''(''t'') denotes the amount of money in a bank account at time ''t'', then the function jumps whenever money is deposited or withdrawn, so the function ''M''(''t'') is discontinuous.
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| ==History==
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| A form of this [[(ε, δ)-definition of limit|epsilon-delta definition]] of continuity was first given by [[Bernard Bolzano]] in 1817. Preliminary forms of a related definition of the limit were given by [[Augustin-Louis Cauchy|Cauchy]].<ref name="grabiner">{{Cite journal|doi=10.2307/2975545|title=Who Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus|first=Judith V.|last=Grabiner|journal=The [[American Mathematical Monthly]]|month=March|year=1983|volume=90|issue=3|pages=185–194|url=http://www.maa.org/pubs/Calc_articles/ma002.pdf|postscript=.|jstor=2975545}}</ref> Cauchy defined continuity of ''f'' as follows: an infinitely small increment of the independent variable ''x'' produces always an infinitely small increment change of ''f(x)''. Cauchy defined infinitely small quantities in terms of variable quantities, and his definition closely parallels the infinitesimal definition used today (see [[microcontinuity]]). The formal definition and the distinction between pointwise continuity and [[uniform continuity]] were first given by Bolzano in the 1830s but the work wasn't published until the 1930s. Heine provided the first published definition of uniform continuity in 1872, but based these ideas on lectures given by Dirichlet in 1854.<ref>{{citation|last1=Rusnock|first1=P.|last2=Kerr-Lawson|first2=A.|title=Bolzano and uniform continuity|journal=Historia Mathematica|volume=32|year=2005|pages=303–311|issue=3}}</ref>
| | :<math forcemathmode="mathml">E=mc^2</math> |
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| == Real-valued continuous functions ==
| | <!--'''PNG''' (currently default in production) |
| ===Definition=== | | :<math forcemathmode="png">E=mc^2</math> |
| 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]]; the function is continuous if, roughly speaking, the graph is a single unbroken [[curve]] with no "holes" or "jumps".
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| 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,
| | '''source''' |
| :<math>f: I \rightarrow \mathbf R.</math>
| | :<math forcemathmode="source">E=mc^2</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 (mathematics)|domain]] of ''f''. Possible choices include ''I''='''R''', the whole set of real numbers, an [[open interval]]
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| :<math>I = (a, b) = \{x \in \mathbf R | a < x < b \}, </math> | |
| or a [[closed interval]]
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| :<math>I = [a, b] = \{x \in \mathbf R | a \leq x \leq b \}. </math>
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| Here, ''a'' and ''b'' are real numbers.
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| ====Definition in terms of limits of functions==== | | <span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples]. |
| The function ''f'' is ''continuous at some [[point (geometry)|point]]'' ''c'' of its domain if the [[limit of a function|limit]] of ''f''(''x'') as ''x'' approaches ''c'' through the domain of ''f'' exists and is equal to ''f''(''c'').<ref>{{Citation | last1=Lang | first1=Serge | author1-link=Serge Lang | title=Undergraduate analysis | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=2nd | series=Undergraduate Texts in Mathematics | isbn=978-0-387-94841-6 | year=1997}}, section II.4</ref> In mathematical notation, this is written as
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| :<math>\lim_{x \to c}{f(x)} = f(c).</math> | |
| In detail this means three conditions: first, ''f'' has to be defined at ''c''. Second, the limit on the left hand side of that equation has to exist. Third, the value of this limit must equal ''f''(''c'').
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| The function ''f'' is said to be ''continuous'' if it is continuous at every point of its domain.
| | ==Demos== |
| If the point ''c'' in the domain of ''f'' is not a [[limit point]] of the domain, then this condition is [[vacuous truth|vacuously true]], since ''x'' cannot approach ''c'' through values not equal ''c''. Thus, for example, every function whose domain is the set of all integers is continuous.
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| ====Definition in terms of limits of sequences====
| | Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]: |
| One can instead require that for any [[sequence (mathematics)|sequence]] <math>(x_n)_{n\in\mathbb{N}}</math> of points in the domain which [[convergent sequence|converges]] to ''c'', the corresponding sequence <math>\left(f(x_n)\right)_{n\in \mathbb{N}}</math> converges to ''f''(''c''). In mathematical notation, <math>\forall (x_n)_{n\in\mathbb{N}} \subset I:\lim_{n\to\infty} x_n=c \Rightarrow \lim_{n\to\infty} f(x_n)=f(c)\,.</math>
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| ==== Weierstrass definition (epsilon-delta) of continuous functions====
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| [[File:Example of continuous function.png|right|thumb|Illustration of the ε-δ-definition: for ε=0.5, the value δ=0.5 satisfies the condition of the definition.]]
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| Explicitly including the definition of the limit of a function, we obtain a self-contained definition:
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| Given a function ''f'' as above and an element ''c'' of the domain ''I'', ''ƒ'' is said to be continuous at the point ''c'' if the following holds: For any number ''ε'' > 0, however small, there exists some number ''δ'' > 0 such that for all ''x'' in the domain of ''ƒ'' with ''c'' − ''δ'' < ''x'' < ''c'' + ''δ'', the value of ''ƒ''(''x'') satisfies
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| :<math> f(c) - \varepsilon < f(x) < f(c) + \varepsilon.\,</math> | | * accessibility: |
| | ** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]] |
| | ** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)] |
| | ** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)] |
| | ** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)] |
| | ** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]]. |
| | ** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]]. |
| | ** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode. |
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| Alternatively written, continuity of ''ƒ'' : ''I'' → ''R'' at ''c'' ∈ ''I'' means that for every ''ε'' > 0 there exists a ''δ'' > 0 such that for all ''x'' ∈ ''I'',:
| | ==Test pages == |
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| :<math>| x - c | < \delta \Rightarrow | f(x) - f(c) | < \varepsilon. \, </math> | | To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages: |
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| More intuitively, we can say that if we want to get all the ''ƒ''(''x'') values to stay in some small [[topological neighbourhood|neighborhood]] around ''ƒ''(''c''), we simply need to choose a small enough neighborhood for the ''x'' values around ''c'', and we can do that no matter how small the ''ƒ''(''x'') neighborhood is; ''ƒ'' is then continuous at ''c''.
| | *[[Inputtypes|Inputtypes (private Wikis only)]] |
| | | *[[Url2Image|Url2Image (private Wikis only)]] |
| In modern terms, this is generalized by the definition of continuity of a function with respect to a [[basis (topology)|basis for the topology]], here the [[metric topology]].
| | ==Bug reporting== |
| | | If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de . |
| ==== Definition using oscillation ====
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| [[File:Rapid Oscillation.svg|thumb|The failure of a function to be continuous at a point is quantified by its [[Oscillation (mathematics)|oscillation]].]] | |
| Continuity can also be defined in terms of [[Oscillation (mathematics)|oscillation]]: a function ƒ is continuous at a point ''x''<sub>0</sub> if and only if the oscillation is zero;<ref>''[http://ramanujan.math.trinity.edu/wtrench/texts/TRENCH_REAL_ANALYSIS.PDF Introduction to Real Analysis],'' updated April 2010, William F. Trench, Theorem 3.5.2, p. 172</ref> in symbols, <math>\omega_f(x_0) = 0.</math> A benefit of this definition is that it ''quantifies'' discontinuity: the oscillation gives how ''much'' the function is discontinuous at a point.
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| This definition is useful in [[descriptive set theory]] to study the set of discontinuities and continuous points – the continuous points are the intersection of the sets where the oscillation is less than ''ε'' (hence a [[G-delta set|G<sub>δ</sub> set]]) – and gives a very quick proof of one direction of the [[Lebesgue integrability condition]].<ref>''[http://ramanujan.math.trinity.edu/wtrench/texts/TRENCH_REAL_ANALYSIS.PDF Introduction to Real Analysis],'' updated April 2010, William F. Trench, 3.5 "A More Advanced Look at the Existence of the Proper Riemann Integral", pp. 171–177</ref>
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| The oscillation is equivalent to the ''ε''-''δ'' definition by a simple re-arrangement, and by using a limit ([[lim sup]], [[lim inf]]) to define oscillation: if (at a given point) for a given ''ε''<sub>0</sub> there is no ''δ'' that satisfies the ''ε''-''δ'' definition, then the oscillation is at least ''ε''<sub>0</sub>, and conversely if for every ''ε'' there is a desired ''δ,'' the oscillation is 0. The oscillation definition can be naturally generalized to maps from a topological space to a metric space.
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| ====Definition using the hyperreals==== | |
| [[Cauchy]] defined continuity of a function in the following intuitive terms: an [[infinitesimal]] change in the independent variable corresponds to an infinitesimal change of the dependent variable (see ''Cours d'analyse'', page 34). [[Non-standard analysis]] is a way of making this mathematically rigorous. The real line is augmented by the addition of infinite and infinitesimal numbers to form the [[hyperreal numbers]]. In nonstandard analysis, continuity can be defined as follows.
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| :A function ''ƒ'' from the reals to the reals is continuous if its natural extension to the hyperreals has the property that for real ''x'' and infinitesimal ''dx'', {{nowrap|''ƒ''(''x''+''dx'') − ''ƒ''(''x'')}} is infinitesimal<ref>http://www.math.wisc.edu/~keisler/calc.html</ref>
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| (see [[microcontinuity]]). In other words, an infinitesimal increment of the independent variable corresponds to an infinitesimal change of the dependent variable, giving a modern expression to [[Augustin-Louis Cauchy]]'s definition of continuity.
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| === Examples ===
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| [[File:Brent method example.png|right|thumb|The graph of a [[cubic function]] has no jumps or holes. The function is continuous.]]
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| All [[polynomial|polynomial function]]s, such as
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| :<math>f(x) = x^3 + x^2 - 5x + 11\,</math>
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| (pictured) are continuous. This is a consequence of the fact that, given two continuous functions
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| :<math>f, g: I \rightarrow \mathbf R</math>
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| defined on the same domain ''I'', then the sum ''f'' + ''g'', and the product ''fg'' of the two functions are continuous (on the same domain ''I''). Moreover, the function
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| :<math>\frac f g : \{ x \in I| g(x) \neq 0 \} \rightarrow \mathbf R, x \mapsto \frac{f(x)}{g(x)}</math>
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| is continuous. (The points where ''g''(''x'') is zero have to be discarded for ''f''/''g'' to be defined.) For example, the function (pictured)
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| :<math>f(x) = \frac {2x-1} {x+2}</math>
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| [[File:Homografia.svg|right|thumb|The graph of a [[rational function]]. The function is not defined for ''x''=−2. The vertical and horizontal lines are [[asymptote]]s.]]
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| is defined for all real numbers {{nowrap|''x'' ≠ −2}} and is continuous at every such point. The question of continuity at {{nowrap|''x'' {{=}} −2}} does not arise, since {{nowrap|''x'' {{=}} −2}} is not in the domain of ''f''. There is no continuous function ''F'': '''R''' → '''R''' that agrees with ''f''(''x'') for all {{nowrap|''x'' ≠ −2}}. The function ''g''(''x'') = (sin ''x'')/''x'', defined for all ''x''≠0 is continuous at these points. However, this function ''can'' be extended to a continuous function on all real numbers, namely
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| :<math>
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| G(x) =
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| \begin{cases}
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| \frac {\sin (x)}x & \text{ if }x \ne 0\\
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| 1 & \text{ if }x = 0,
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| \end{cases}
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| </math>
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| since the limit of ''g''(''x''), when ''x'' approaches 0, is 1. Therefore, the point ''x''=0 is called a [[removable singularity]] of ''g''.
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| Given two continuous functions
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| :<math>f: I \rightarrow J (\subset \mathbf R), g : J \rightarrow \mathbf R,</math>
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| the [[Function composition|composition]]
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| :<math>g \circ f : I \rightarrow \mathbf R, x \mapsto g(f(x))</math>
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| is continuous.
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| ===Non-examples===
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| An example of a discontinuous function is the function ''f'' defined by ''f''(''x'') = 1 if ''x'' > 0, ''f''(''x'') = 0 if ''x'' ≤ 0. Pick for instance ε = {{frac|1|2}}. There is no δ-neighborhood around ''x'' = 0 that will force all the ''f''(''x'') values to be within ε of ''f''(0). Intuitively we can think of this type of discontinuity as a sudden jump in function values. Similarly, the [[sign function|signum]] or sign function
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| [[File:Signum function.svg|right|thumb|Plot of the signum function. The hollow dots indicate that sgn(''x'') is 1 for all ''x''>0 and −1 for all ''x''<0.]]
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| :<math>
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| \sgn(x) = \begin{cases}
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| 1 & \text{ if }x > 0\\
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| 0 & \text{ if }x = 0\\
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| -1 & \text{ if }x < 0
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| \end{cases}
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| </math>
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| is discontinuous at ''x'' = 0 but continuous everywhere else. Yet another example: the function
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| :<math>f(x)=\begin{cases}
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| \sin(\frac{1}{x^2})\text{ if }x \ne 0\\
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| 0\text{ if }x = 0
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| \end{cases}</math>
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| is continuous everywhere apart from ''x'' = 0.
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| [[File:Thomae function (0,1).svg|right|thumb|Plot of Thomae's function for the domain {{nowrap|−1 ≤ ''x'' ≤ 1}}.]]
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| [[Thomae's function]],
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| :<math>f(x)=\begin{cases}
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| 1 \text{ if }x=0\\
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| \frac{1}{q}\text{ if }x=\frac{p}{q}\text{(in lowest terms) is a rational number}\\
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| 0\text{ if }x\text{ is irrational}.
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| \end{cases}</math>
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| is continuous at all irrational numbers and discontinuous at all rational numbers. In a similar vein, [[Dirichlet's function]]
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| :<math>D(x)=\begin{cases}
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| 0\text{ if }x \in \mathbb{R} \setminus \mathbb{Q}\\
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| 1\text{ if }x \in \mathbb{Q}
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| \end{cases}</math>
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| is nowhere continuous.
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| ===Properties===
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| ====Intermediate value theorem====
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| The [[intermediate value theorem]] is an [[existence theorem]], based on the real number property of [[Real number#Completeness|completeness]], and states:
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| : If the real-valued function ''f'' is continuous on the [[interval (mathematics)|closed interval]] [''a'', ''b''] and ''k'' is some number between ''f''(''a'') and ''f''(''b''), then there is some number ''c'' in [''a'', ''b''] such that ''f''(''c'') = ''k''.
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| For example, if a child grows from 1 m to 1.5 m between the ages of two and six years, then, at some time between two and six years of age, the child's height must have been 1.25 m.
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| As a consequence, if ''f'' is continuous on [''a'', ''b''] and ''f''(''a'') and ''f''(''b'') differ in [[Sign (mathematics)|sign]], then, at some point ''c'' in [''a'', ''b''], ''f''(''c'') must equal [[0 (number)|zero]].
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| ====Extreme value theorem====
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| The [[extreme value theorem]] states that if a function ''f'' is defined on a closed interval [''a'',''b''] (or any closed and bounded set) and is continuous there, then the function attains its maximum, i.e. there exists ''c'' ∈ [''a'',''b''] with ''f''(''c'') ≥ ''f''(''x'') for all ''x'' ∈ [''a'',''b'']. The same is true of the minimum of ''f''. These statements are not, in general, true if the function is defined on an open interval (''a'',''b'') (or any set that is not both closed and bounded), as, for example, the continuous function ''f''(''x'') = 1/''x'', defined on the open interval (0,1), does not attain a maximum, being unbounded above.
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| ====Relation to differentiability and integrability====
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| Every [[differentiable function]]
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| :<math>f: (a, b) \rightarrow \mathbf R</math>
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| is continuous, as can be shown. The [[Theorem#Converse|converse]] does not hold: for example, the [[absolute value]] function
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| :<math>f(x)=|x| = \begin{cases}
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| x \text{ if }x \geq 0\\
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| -x\text{ if }x < 0
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| \end{cases}</math>
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| is everywhere continuous. However, it is not differentiable at ''x'' = 0 (but is so everywhere else). [[Weierstrass function|Weierstrass's function]] is everywhere continuous but nowhere differentiable.
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| The [[derivative]] ''f''<nowiki>'</nowiki>(''x'') of a differentiable function ''f''(''x'') need not be continuous. If ''f''<nowiki>'</nowiki>(''x'') is continuous, ''f''(''x'') is said to be continuously differentiable. The set of such functions is denoted ''C''<sup>1</sup>((''a'', ''b'')). More generally, the set of functions
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| :<math>f: \Omega \rightarrow \mathbf R</math>
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| (from an open interval (or [[open subset]] of '''R''') Ω to the reals) such that ''f'' is ''n'' times differentiable and such that the ''n''-th derivative of ''f'' is continuous is denoted ''C''<sup>''n''</sup>(Ω). See [[differentiability class]]. In the field of computer graphics, these three levels are sometimes called ''G''<sup>0</sup> (continuity of position), ''G''<sup>1</sup> (continuity of tangency), and ''G''<sup>2</sup> (continuity of curvature).
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| Every continuous function
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| :<math>f: [a, b] \rightarrow \mathbf R</math>
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| is [[integrable function|integrable]] (for example in the sense of the [[Riemann integral]]). The converse does not hold, as the (integrable, but discontinuous) [[sign function]] shows.
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| ====Pointwise and uniform limits====
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| [[File:Uniform continuity animation.gif|A sequence of continuous functions ''f''<sub>''n''</sub>(''x'') whose (pointwise) limit function ''f''(''x'') is discontinuous. The convergence is not uniform.|right|thumb]]
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| Given a [[sequence (mathematics)|sequence]]
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| :<math>f_1, f_2, \dots : I \rightarrow \mathbf R</math>
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| of functions such that the limit
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| :<math>f(x) := \lim_{n \rightarrow \infty} f_n(x)</math>
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| exists for all ''x'' in ''I'', the resulting function ''f''(''x'') is referred to as the [[pointwise convergence|pointwise limit]] of the sequence of functions (''f''<sub>''n''</sub>)<sub>''n''∈'''N'''</sub>. The pointwise limit function need not be continuous, even if all functions ''f''<sub>''n''</sub> are continuous, as the animation at the right shows. However, ''f'' is continuous when the sequence [[uniform convergence|converges uniformly]], by the [[uniform convergence theorem]]. This theorem can be used to show that the [[exponential function]]s, [[logarithm]]s, [[square root]] function, [[trigonometric function]]s are continuous.
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| ===Directional and semi-continuity===
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| <div style="float:right;">
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| <gallery>Image:Right-continuous.svg|A right-continuous function
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| Image:Left-continuous.svg|A left-continuous function</gallery></div>
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| Discontinuous functions may be discontinuous in a restricted way, giving rise to the concept of directional continuity (or right and left continuous functions) and [[semi-continuity]]. Roughly speaking, a function is ''right-continuous'' if no jump occurs when the limit point is approached from the right. More formally, ''ƒ'' is said to be right-continuous at the point ''c'' if the following holds: For any number ''ε'' > 0 however small, there exists some number ''δ'' > 0 such that for all ''x'' in the domain with {{nowrap|''c'' < ''x'' < ''c'' + ''δ''}}, the value of ''ƒ''(''x'') will satisfy
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| :<math> |f(x) - f(c)| < \varepsilon.\,</math>
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| This is the same condition as for continuous functions, except that it is required to hold for ''x'' strictly larger than ''c'' only. Requiring it instead for all ''x'' with {{nowrap|''c'' − ''δ'' < ''x'' < ''c''}} yields the notion of ''left-continuous'' functions. A function is continuous if and only if it is both right-continuous and left-continuous.
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| A function ''f'' is ''[[Semi-continuity|lower semi-continuous]]'' if, roughly, any jumps that might occur only go down, but not up. That is, for any ''ε'' > 0, there exists some number ''δ'' > 0 such that for all ''x'' in the domain with {{nowrap begin}}|x − c| < ''δ''{{nowrap end}}, the value of ''ƒ''(''x'') satisfies
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| :<math>f(x) \geq f(c) - \epsilon.</math>
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| The reverse condition is ''[[Semi-continuity|upper semi-continuity]]''.
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| == Continuous functions between metric spaces ==
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| <!-- This section is linked from [[F-space]] -->
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| The concept of continuous real-valued functions can be generalized to functions between [[metric space]]s. A metric space is a set ''X'' equipped with a function (called [[metric (mathematics)|metric]]) ''d''<sub>''X''</sub>, that can be thought of as a measurement of the distance of any two elements in ''X''. Formally, the metric is a function
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| :<math>d_X : X \times X \rightarrow \mathbf R</math>
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| that satisfies a number of requirements, notably the [[triangle inequality]]. Given two metric spaces (''X'', d<sub>''X''</sub>) and (''Y'', d<sub>''Y''</sub>) and a function
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| :<math>f: X \rightarrow Y</math>
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| then ''f'' is continuous at the point ''c'' in ''X'' (with respect to the given metrics) if for any positive real number ε, there exists a positive real number δ such that all ''x'' in ''X'' satisfying d<sub>''X''</sub>(''x'', ''c'') < δ will also satisfy d<sub>''Y''</sub>(''f''(''x''), ''f''(''c'')) < ε. As in the case of real functions above, this is equivalent to the condition that for every sequence (''x''<sub>''n''</sub>) in ''X'' with limit lim ''x''<sub>''n''</sub> = ''c'', we have lim ''f''(''x''<sub>''n''</sub>) = ''f''(''c''). The latter condition can be weakened as follows: ''f'' is continuous at the point ''c'' if and only if for every convergent sequence (''x''<sub>''n''</sub>) in ''X'' with limit ''c'', the sequence (''f''(''x''<sub>''n''</sub>)) is a [[Cauchy sequence]], and ''c'' is in the domain of ''f''.
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| The set of points at which a function between metric spaces is continuous is a [[Gδ set|G<sub>δ</sub> set]] – this follows from the ε-δ definition of continuity.
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| This notion of continuity is applied, for example, in [[functional analysis]]. A key statement in this area says that a [[linear operator]]
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| :<math>T: V \rightarrow W</math>
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| between [[normed vector space]]s ''V'' and ''W'' (which are [[vector spaces]] equipped with a compatible [[norm (mathematics)|norm]], denoted ||x||)
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| is continuous if and only if it is [[Bounded linear operator|bounded]], that is, there is a constant ''K'' such that
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| :<math>\|T(x)\| \leq K \|x\|</math>
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| for all ''x'' in ''V''.
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| === Uniform, Hölder and Lipschitz continuity ===
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| [[Image:Lipschitz continuity.png|thumb|For a Lipschitz continuous function, there is a double cone (shown in white) whose vertex can be translated along the graph, so that the graph always remains entirely outside the cone.]]
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| The concept of continuity for functions between metric spaces can be strengthened in various ways by limiting the way δ depends on ε and ''c'' in the definition above. Intuitively, a function ''f'' as above is [[uniformly continuous]] if the δ does
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| not depend on the point ''c''. More precisely, it is required that for every [[real number]] ''ε'' > 0 there exists ''δ'' > 0 such that for every ''c'', ''b'' ∈ ''X'' with ''d''<sub>''X''</sub>(''b'', ''c'') < ''δ'', we have that ''d''<sub>''Y''</sub>(''f''(''b''), ''f''(''c'')) < ''ε''. Thus, any uniformly continuous function is continuous. The converse does not hold in general, but holds when the domain space ''X'' is [[compact topological space|compact]]. Uniformly continuous maps can be defined in the more general situation of [[uniform space]]s.<ref>{{Citation | last1=Gaal | first1=Steven A. | title=Point set topology | publisher=[[Dover Publications]] | location=New York | isbn=978-0-486-47222-5 | year=2009}}, section IV.10</ref>
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| A function is [[Hölder continuity|Hölder continuous]] with exponent α (a real number) if there is a constant ''K'' such that for all ''b'' and ''c'' in ''X'', the inequality
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| :<math>d_Y (f(b), f(c)) \leq K \cdot (d_X (b, c))^\alpha</math>
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| holds. Any Hölder continuous function is uniformly continuous. The particular case {{nowrap|α {{=}} 1}} is referred to as [[Lipschitz continuity]]. That is, a function is Lipschitz continuous if there is a constant ''K'' such that the inequality
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| :<math>d_Y (f(b), f(c)) \leq K \cdot d_X (b, c)</math>
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| holds any ''b'', ''c'' in ''X''.<ref>{{Citation | last1=Searcóid | first1=Mícheál Ó | title=Metric spaces | url=http://books.google.de/books?id=aP37I4QWFRcC | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Springer undergraduate mathematics series | isbn=978-1-84628-369-7 | year=2006}}, section 9.4</ref> The Lipschitz condition occurs, for example, in the [[Picard–Lindelöf theorem]] concerning the solutions of [[ordinary differential equation]]s.
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| == Continuous functions between topological spaces<!--This section is linked from [[Preference (economics)]]--> ==
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| [[Image:continuity topology.svg|300px|right|frame|Continuity of a function at a point]]
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| Another, more abstract notion of continuity is continuity of functions between [[topological space]]s in which there generally is no formal notion of distance, as in the case of metric spaces. A topological space is a set ''X'' together with a [[topology]] on ''X'' which is a set of [[subset]]s of ''X'' satisfying a few requirements with respect to their unions and intersections that generalize the properties of the [[open ball]]s in metric spaces while still allowing to talk about the [[neighbourhood (mathematics)|neighbourhoods]] of a given point. The elements of a topology are called [[open subset]]s of ''X'' (with respect to the topology).
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| A function
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| :{{nowrap|''f'' : ''X'' → ''Y''}}
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| between two topological spaces ''X'' and ''Y'' is continuous if for every open set ''V'' ⊆ ''Y'', the inverse image
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| :<math>f^{-1}(V) = \{x \in X \; | \; f(x) \in V \}</math>
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| is an open subset of ''X''. That is, ''f'' is a function between the sets ''X'' and ''Y'' (not on the elements of the topology ''T<sub>X</sub>''), but the continuity of ''f'' depends on the topologies used on ''X'' and ''Y''.
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| This is equivalent to the condition that the preimages of the [[closed set]]s (which are the complements of the open subsets) in ''Y'' are closed in ''X''.
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| An extreme example: if a set ''X'' is given the [[discrete topology]] (in which every subset is open), all functions
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| :<math>f: X \rightarrow T</math>
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| to any topological space ''T'' are continuous. On the other hand, if ''X'' is equipped with the [[indiscrete topology]] (in which the only open subsets are the empty set and ''X'') and the space ''T'' set is at least [[T0 space|T<sub>0</sub>]], then the only continuous functions are the constant functions. Conversely, any function whose range is indiscrete is continuous.
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| === Alternative definitions ===
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| Several [[Characterizations of the category of topological spaces|equivalent definitions for a topological structure]] exist and thus there are several equivalent ways to define a continuous function.
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| ==== Neighborhood definition ====
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| Definitions based on preimages are often difficult to use directly. The following criterion expresses continuity in terms of [[neighborhood (topology)|neighborhood]]s: ''f'' is continuous at some point ''x'' ∈ ''X'' if and only if for any neighborhood ''V'' of ''f''(''x''), there is a neighborhood ''U'' of ''x'' such that ''f''(''U'') ⊆ ''V''. Intuitively, continuity means no matter how "small" ''V'' becomes, there is always a ''U'' containing ''x'' that maps inside ''V''.
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| If ''X'' and ''Y'' are metric spaces, it is equivalent to consider the [[neighborhood system]] of [[open ball]]s centered at ''x'' and ''f''(''x'') instead of all neighborhoods. This gives back the above δ-ε definition of continuity in the context of metric spaces. However, in general topological spaces, there is no notion of nearness or distance.
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| Note, however, that if the target space is [[Hausdorff space|Hausdorff]], it is still true that ''f'' is continuous at ''a'' if and only if the limit of ''f'' as ''x'' approaches ''a'' is ''f(a)''. At an isolated point, every function is continuous.
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| ==== Sequences and nets {{anchor|Heine definition of continuity}} ====
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| In several contexts, the topology of a space is conveniently specified in terms of [[limit points]]. In many instances, this is accomplished by specifying when a point is the [[limit of a sequence]], but for some spaces that are too large in some sense, one specifies also when a point is the limit of more general sets of points indexed by a [[directed set]], known as [[net (mathematics)|nets]]. A function is continuous only if it takes limits of sequences to limits of sequences. In the former case, preservation of limits is also sufficient; in the latter, a function may preserve all limits of sequences yet still fail to be continuous, and preservation of nets is a necessary and sufficient condition.
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| In detail, a function ''f'' : ''X'' → ''Y'' is '''sequentially continuous''' if whenever a sequence (''x''<sub>''n''</sub>) in ''X'' converges to a limit ''x'', the sequence (''f''(''x''<sub>''n''</sub>)) converges to ''f''(''x''). Thus sequentially continuous functions "preserve sequential limits". Every continuous function is sequentially continuous. If ''X'' is a [[first-countable space]] and [[Axiom of countable choice|countable choice]] holds, then the converse also holds: any function preserving sequential limits is continuous. In particular, if ''X'' is a metric space, sequential continuity and continuity are equivalent. For non first-countable spaces, sequential continuity might be strictly weaker than continuity. (The spaces for which the two properties are equivalent are called [[sequential space]]s.) This motivates the consideration of nets instead of sequences in general topological spaces. Continuous functions preserve limits of nets, and in fact this property characterizes continuous functions.
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| ==== Closure operator definition ====
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| Instead of specifying the open subsets of a topological space, the topology can also be determined by [[closure operator]]s (denoted cl) which assigns to any subset ''A'' ⊆ ''X'' its [[closure]] or [[interior operator]]s (denoted int), which assigns to any subset ''A'' of ''X'' its [[interior]]. In these terms, a function
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| :<math>f:(X,\mathrm{cl}) \to (X' ,\mathrm{cl}')\, </math>
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| between topological spaces is continuous in the sense above if and only if for all subsets ''A'' of ''X''
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| :<math>f(\mathrm{cl}(A)) \subseteq \mathrm{cl}'(f(A)).</math>
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| That is to say, given any element ''x'' of ''X'' that is in the closure of any subset ''A'', ''f''(''x'') belongs to the closure of ''f''(''A''). This is equivalent to the requirement that for all subsets ''A''<nowiki>'</nowiki> of ''X''<nowiki>'</nowiki>
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| :<math>f^{-1}(\mathrm{cl}'(A')) \supseteq \mathrm{cl}(f^{-1}(A')).</math>
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| Moreover,
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| :<math>f:(X,\mathrm{int}) \to (X' ,\mathrm{int}') \, </math>
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| is continuous if and only if
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| :<math>f^{-1}(\mathrm{int}'(A)) \subseteq \mathrm{int}(f^{-1}(A)).</math>
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| for any subset ''A'' of ''X''.
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| === Properties ===
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| If ''f'' : ''X'' → ''Y'' and ''g'' : ''Y'' → ''Z'' are continuous, then so is the composition ''g'' ∘ ''f'' : ''X'' → ''Z''. If ''f'' : ''X'' → ''Y'' is continuous and
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| * ''X'' is [[Compact space|compact]], then ''f''(''X'') is compact.
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| * ''X'' is [[Connected space|connected]], then ''f''(''X'') is connected.
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| * ''X'' is [[path-connected]], then ''f''(''X'') is path-connected.
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| * ''X'' is [[Lindelöf space|Lindelöf]], then ''f''(''X'') is Lindelöf.
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| * ''X'' is [[separable space|separable]], then ''f''(''X'') is separable.
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| The possible topologies on a fixed set ''X'' are [[partial ordering|partially ordered]]: a topology τ<sub>1</sub> is said to be [[comparison of topologies|coarser]] than another topology τ<sub>2</sub> (notation: τ<sub>1</sub> ⊆ τ<sub>2</sub>) if every open subset with respect to τ<sub>1</sub> is also open with respect to τ<sub>2</sub>. Then, the [[identity function|identity map]]
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| :id<sub>X</sub> : (''X'', τ<sub>2</sub>) → (''X'', τ<sub>1</sub>)
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| is continuous if and only if τ<sub>1</sub> ⊆ τ<sub>2</sub> (see also [[comparison of topologies]]). More generally, a continuous function
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| :<math>(X, \tau_X) \rightarrow (Y, \tau_Y)</math>
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| stays continuous if the topology τ<sub>''X''</sub> is replaced by a [[weaker topology]] and/or τ<sub>''Y''</sub> is replaced by a [[stronger topology]].
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| ===Homeomorphisms===
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| Symmetric to the concept of a continuous map is an [[open map]], for which ''images'' of open sets are open. In fact, if an open map ''f'' has an [[inverse function]], that inverse is continuous, and if a continuous map ''g'' has an inverse, that inverse is open. Given a bijective function ''f'' between two topological spaces, the inverse function ''f''<sup>−1</sup> need not be continuous. A bijective continuous function with continuous inverse function is called a ''[[homeomorphism]]''.
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| If a continuous bijection has as its domain a [[compact space]] and its codomain is [[Hausdorff space|Hausdorff]], then it is a homeomorphism.
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| === Defining topologies via continuous functions ===
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| Given a function
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| :<math>f: X \rightarrow S, \,</math>
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| where ''X'' is a topological space and ''S'' is a set (without a specified topology), the [[final topology]] on ''S'' is defined by letting the open sets of ''S'' be those subsets ''A'' of ''S'' for which ''f''<sup>−1</sup>(''A'') is open in ''X''. If ''S'' has an existing topology, ''f'' is continuous with respect to this topology if and only if the existing topology is [[Comparison of topologies|coarser]] than the final topology on ''S''. Thus the final topology can be characterized as the finest topology on ''S'' that makes ''f'' continuous. If ''f'' is [[surjective]], this topology is canonically identified with the [[quotient topology]] under the [[equivalence relation]] defined by ''f''.
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| Dually, for a function ''f'' from a set ''S'' to a topological space, the [[initial topology]] on ''S'' has as open subsets ''A'' of ''S'' those subsets for which ''f''(''A'') is open in ''X''. If ''S'' has an existing topology, ''f'' is continuous with respect to this topology if and only if the existing topology is finer than the initial topology on ''S''. Thus the initial topology can be characterized as the coarsest topology on ''S'' that makes ''f'' continuous. If ''f'' is injective, this topology is canonically identified with the [[subspace topology]] of ''S'', viewed as a subset of ''X''.
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| More generally, given a set ''S'', specifying the set of continuous functions
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| :<math>S \rightarrow X</math>
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| into all topological spaces ''X'' defines a topology. [[Duality (mathematics)|Dually]], a similar idea can be applied to maps
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| :<math>X \rightarrow S.</math>
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| This is an instance of a [[universal property]].
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| ==Related notions==
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| Various other mathematical domains use the concept of continuity in different, but related meanings. For example, in [[order theory]], an order-preserving function ''f'' : ''X'' → ''Y'' between two [[complete lattice]]s ''X'' and ''Y'' (particular types of [[partially ordered set]]s) is [[Scott continuity|continuous]] if for each subset ''A'' of ''X'', we have sup(''f''(''A'')) = ''f''(sup(''A'')). Here sup is the [[supremum]] with respect to the orderings in ''X'' and ''Y'', respectively. Applying this to the complete lattice consisting of the open subsets of a topological space, this gives back the notion of continuity for maps between topological spaces.
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| In [[category theory]], a [[functor]]
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| :<math>F : \mathcal C \rightarrow \mathcal D</math>
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| between two [[category (mathematics)|categories]] is called ''[[continuous functor|continuous]]'', if it commutes with small [[limit (category theory)|limits]]. That is to say,
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| :<math>\varprojlim_{i \in I} F(C_i) \cong F (\varprojlim_{i \in I} C_i)</math>
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| for any small (i.e., indexed by a set ''I'', as opposed to a [[class (mathematics)|class]]) diagram of objects in <math>\mathcal C</math>.
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| A ''continuity space''<ref>{{cite journal | title = Quantales and continuity spaces | id = {{citeseerx|10.1.1.48.851}} | first = R. C. | last =Flagg | journal = Algebra Universalis | year = 1997 }}</ref><ref>{{cite journal | title = All topologies come from generalized metrics | first = R. | last = Kopperman | journal = American Mathematical Monthly | year = 1988 }}</ref> is a generalization of metric spaces and posets, which uses the concept of [[quantale]]s, and that can be used to unify the notions of metric spaces and [[Domain theory|domain]]s.<ref>{{cite journal | title = Continuity spaces: Reconciling domains and metric spaces | first1 = B. | last1 = Flagg | first2 = R. | last2 = Kopperman | journal = Theoretical Computer Science | year = 1997 }}</ref>
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| == See also ==
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| <div style="-moz-column-count:2; column-count:2;">
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| * [[Absolute continuity]]
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| * [[Classification of discontinuities]]
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| * [[Coarse function]]
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| * [[Continuous stochastic process]]
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| * [[Dini continuity]]
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| * [[Discrete function]]
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| * [[Equicontinuity]]
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| * [[Normal function]]
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| * [[Piecewise]]
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| * [[Symmetrically continuous function]]
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| </div>
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| ==Notes==
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| {{commons category|Continuity (functions)}}
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| {{Reflist}}
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| ==References==
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| *[http://archives.math.utk.edu/visual.calculus/ Visual Calculus] by Lawrence S. Husch, [[University of Tennessee]] (2001)
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| {{DEFAULTSORT:Continuous Function}}
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| [[Category:Calculus]]
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| [[Category:Continuous mappings| ]]
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| [[Category:Types of functions]]
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