Editing Continuous function (section)

==Continuous functions between metric spaces== <!--This section is linked from [[F-space]]-->
{{anchor|Metric spaces}}

The concept of continuous real-valued functions can be generalized to functions between [[metric space]]s. A metric space is a set <math>X</math> equipped with a function (called [[Metric (mathematics)|metric]]) <math>d_X,</math> that can be thought of as a measurement of the distance of any two elements in ''X''. Formally, the metric is a function
<math display="block">d_X : X \times X \to \R</math>
that satisfies a number of requirements, notably the [[triangle inequality]]. Given two metric spaces <math>\left(X, d_X\right)</math> and <math>\left(Y, d_Y\right)</math> and a function
<math display="block">f : X \to Y</math>
then <math>f</math> is continuous at the point <math>c \in X</math> (with respect to the given metrics) if for any positive real number <math>\varepsilon > 0,</math> there exists a positive real number <math>\delta > 0</math> such that all <math>x \in X</math> satisfying <math>d_X(x, c) < \delta</math> will also satisfy <math>d_Y(f(x), f(c)) < \varepsilon.</math> As in the case of real functions above, this is equivalent to the condition that for every sequence <math>\left(x_n\right)</math> in <math>X</math> with limit <math>\lim x_n = c,</math> we have <math>\lim f\left(x_n\right) = f(c).</math> The latter condition can be weakened as follows: <math>f</math> is continuous at the point <math>c</math> if and only if for every convergent sequence <math>\left(x_n\right)</math> in <math>X</math> with limit <math>c</math>, the sequence <math>\left(f\left(x_n\right)\right)</math> is a [[Cauchy sequence]], and <math>c</math> is in the domain of <math>f</math>.

The set of points at which a function between metric spaces is continuous is a [[Gδ set|<math>G_{\delta}</math> set]]&nbsp;– this follows from the <math>\varepsilon-\delta</math> definition of continuity.

This notion of continuity is applied, for example, in [[functional analysis]]. A key statement in this area says that a [[linear operator]]
<math display="block">T : V \to W</math>
between [[normed vector space]]s <math>V</math> and <math>W</math> (which are [[vector space]]s equipped with a compatible [[norm (mathematics)|norm]], denoted <math>\|x\|</math>) is continuous if and only if it is [[Bounded linear operator|bounded]], that is, there is a constant <math>K</math> such that
<math display="block">\|T(x)\| \leq K \|x\|</math>
for all <math>x \in V.</math>

===Uniform, Hölder and Lipschitz continuity===
[[File: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.]]
The concept of continuity for functions between metric spaces can be strengthened in various ways by limiting the way <math>\delta</math> depends on <math>\varepsilon</math> and ''c'' in the definition above.  Intuitively, a function ''f'' as above is [[uniformly continuous]] if the <math>\delta</math> does
not depend on the point ''c''. More precisely, it is required that for every [[real number]] <math>\varepsilon > 0</math> there exists <math>\delta > 0</math> such that for every <math>c, b \in X</math> with <math>d_X(b, c) < \delta,</math> we have that <math>d_Y(f(b), f(c)) < \varepsilon.</math> Thus, any uniformly continuous function is continuous. The converse does not generally hold 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>

A function is [[Hölder continuity|Hölder continuous]] with exponent α (a real number) if there is a constant ''K'' such that for all <math>b, c \in X,</math> the inequality
<math display="block">d_Y (f(b), f(c)) \leq K \cdot (d_X (b, c))^\alpha</math>
holds. Any Hölder continuous function is uniformly continuous. The particular case <math>\alpha = 1</math> is referred to as [[Lipschitz continuity]]. That is, a function is Lipschitz continuous if there is a constant ''K'' such that the inequality
<math display="block">d_Y (f(b), f(c)) \leq K \cdot d_X (b, c)</math>
holds for any <math>b, c \in X.</math><ref>{{Citation | last1=Searcóid | first1=Mícheál Ó | title=Metric spaces | url=https://books.google.com/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.