Editing Uniform continuity (section)

== Other characterizations ==

=== Non-standard analysis ===
In [[non-standard analysis]], a real-valued function ''<math>f</math>'' of a real variable is [[microcontinuity|microcontinuous]] at a point ''<math>a</math>'' precisely if the difference <math>f^*(a + \delta) - f^*(a)</math> is infinitesimal whenever ''<math>\delta</math>'' is infinitesimal. Thus ''<math>f</math>'' is continuous on a set ''<math>A</math>'' in <math>\mathbb{R}</math> precisely if <math>f^*</math> is microcontinuous at every real point <math>a \in A</math>. Uniform continuity can be expressed as the condition that (the natural extension of) <math>f</math> is microcontinuous not only at real points in <math>A</math>, but at all points in its non-standard counterpart (natural extension) <math>^*A</math> in <math>^*\mathbb{R}</math>. Note that there exist hyperreal-valued functions which meet this criterion but are not uniformly continuous, as well as uniformly continuous hyperreal-valued functions which do not meet this criterion, however, such functions cannot be expressed in the form <math>f^*</math> for any real-valued function <math>f</math>. (see [[non-standard calculus]] for more details and examples).

=== Cauchy continuity ===

For a function between metric spaces, uniform continuity implies [[Cauchy continuity]] {{harv|Fitzpatrick|2006}}. More specifically, let <math>A</math> be a subset of <math>\mathbb{R}^n</math>. If a function <math>f:A \to \mathbb{R}^n</math> is uniformly continuous then for every pair of sequences <math>x_n</math> and <math>y_n</math> such that

:<math>\lim_{n\to\infty} |x_n-y_n|=0</math>

we have

:<math>\lim_{n\to\infty} |f(x_n)-f(y_n)|=0.</math>