Editing Continuous function (section)

===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.