Editing Uniform continuity (section)

== Properties ==

Every uniformly continuous function is [[continuous function|continuous]], but the converse does not hold. Consider for instance the continuous function <math>f \colon \mathbb{R} \rightarrow \mathbb{R}, x
\mapsto x^2</math> where <math>\mathbb{R}</math> is [[Real number|the set of real numbers]]. Given a positive real number <math>\varepsilon</math>, uniform continuity requires the existence of a positive real number <math>\delta</math> such that for all <math>x_1, x_2 \in \mathbb{R}</math> with <math>|x_1 - x_2| < \delta</math>, we have <math>|f(x_1)-f(x_2)| < \varepsilon</math>. But
: <math>f\left(x + \delta \right)-f(x) = 2x\cdot \delta + \delta^2,</math>
and as <math>x</math> goes to be a higher and higher value, <math>\delta</math> needs to be lower and lower to satisfy <math>|f(x + \beta) -f(x)| < \varepsilon</math> for positive real numbers <math>\beta < \delta</math> and the given <math>\varepsilon</math>. This means that there is no specifiable (no matter how small it is) positive real number <math>\delta</math> to satisfy the condition for <math>f</math> to be uniformly continuous so <math>f</math> is not uniformly continuous.

Any [[absolutely continuous]] function (over a compact interval) is uniformly continuous. On the other hand, the [[Cantor function]] is uniformly continuous but not absolutely continuous.

The image of a [[totally bounded space|totally bounded]] subset under a uniformly continuous function is totally bounded. However, the image of a bounded subset of an arbitrary metric space under a uniformly continuous function need not be bounded: as a counterexample, consider the identity function from the integers endowed with the [[discrete metric]] to the integers endowed with the usual [[Euclidean metric]].

The [[Heine–Cantor theorem]] asserts that ''every continuous function on a [[compact set]] is uniformly continuous''. In particular, ''if a function is continuous on a [[interval (mathematics)|closed bounded interval]] of the real line, it is uniformly continuous on that interval''. The [[Darboux integral|Darboux integrability]] of continuous functions follows almost immediately from this theorem.

If a real-valued function <math>f</math> is continuous on <math>[0, \infty)</math> and <math>\lim_{x \to \infty} f(x)</math> exists (and is finite), then <math>f</math> is uniformly continuous. In particular, every element of <math>C_0(\mathbb{R})</math>, the space of continuous functions on <math>\mathbb{R}</math> that vanish at infinity, is uniformly continuous.<!-- This is true for some more general X instead of R, but I don't know how general X can get. --> This is a generalization of the Heine-Cantor theorem mentioned above, since <math>C_c(\mathbb{R}) \subset C_0(\mathbb{R}) </math>.