Editing Uniform continuity (section)

{{Short description|Uniform restraint of the change in functions}}[[File:Continuity and uniform continuity 2.gif|thumb|upright=1.5|As the center of the blue window, with real height <math>2 \varepsilon\in\mathbb R_{>0}</math> and real width <math>2\delta\in\mathbb R_{>0}</math>, moves over the graph of <math>f(x)=\tfrac1x</math> in the direction of <math> x=0</math>, there comes a point at which the graph of <math>f</math> penetrates the (interior of the) top and/or bottom of that window. This means that <math>f</math> ranges over an interval larger than or equal to <math>\varepsilon</math> over an <math>x</math>-interval smaller than <math>\delta</math>. If there existed a window whereof top and/or bottom is never penetrated by the graph of <math>f</math> as the window moves along it over its domain, then that window's width would need to be infinitesimally small (nonreal), meaning that <math>f(x)</math> is '''not''' uniformly continuous. The function <math>g(x)=\sqrt x</math>, on the other hand, '''is''' uniformly continuous.]]
In [[mathematics]], a real [[function (mathematics)|function]] <math>f</math> of real numbers is said to be '''uniformly continuous''' if there is a positive real number <math>\delta</math> such that function values over any function domain interval of the size <math>\delta</math> are as close to each other as we want. In other words, for a uniformly continuous real function of real numbers, if we want function value differences to be less than any positive real number <math>\varepsilon</math>, then there is a positive real number <math>\delta</math> such that <math>|f(x) - f(y)| < \varepsilon</math> for any <math>x</math> and <math>y</math> in any interval of length <math>\delta</math> within the domain of <math>f</math>. 

The difference between uniform continuity and (ordinary) [[Continuous function|continuity]] is that in uniform continuity there is a globally applicable <math>\delta</math> (the size of a function domain interval over which function value differences are less than <math>\varepsilon</math>) that depends on only <math>\varepsilon</math>, while in (ordinary) continuity there is a locally applicable <math>\delta</math> that depends on both <math>\varepsilon</math> and <math>x</math>. So uniform continuity is a stronger continuity condition than continuity; a function that is uniformly continuous is continuous but a function that is continuous is not necessarily uniformly continuous. The concepts of uniform continuity and continuity can be expanded to functions defined between [[metric spaces]]. 

Continuous functions can fail to be uniformly continuous if they are unbounded on a bounded domain, such as <math>f(x) = \tfrac1x</math> on <math> (0,1) </math>, or if their slopes become unbounded on an infinite domain, such as <math>f(x)=x^2</math> on the real (number) line. However, any [[Lipschitz continuity|Lipschitz map]] between metric spaces is uniformly continuous, in particular any [[isometry]] (distance-preserving map). 

Although continuity can be defined for functions between general topological spaces, defining uniform continuity requires more structure. The concept relies on comparing the sizes of [[neighbourhood (mathematics)|neighbourhood]]s of distinct points, so it requires a metric space, or more generally a [[uniform space]].