uniformly continuous

Let $f: A \rightarrow \mathbb{R}$ be a real function defined on a subset $A$ of the real line. We say that $f$ is uniformly continuous if, given an arbitrary small positive $\varepsilon$ , there exists a positive $\delta$ such that whenever two points in $A$ differ by less than $\delta$ , they are mapped by $f$ into points which differ by less than $\varepsilon$ . In symbols:$$ \forall \varepsilon > 0\ \exists \delta > 0\ \forall x,y \in A\ |x-y| < \delta \Rightarrow |f(x)-f(y)| < \varepsilon.$$

Every uniformly continuous function is also continuous, while the converse does not always hold. For instance, the function $f: ]0,+\infty[ \rightarrow \mathbb{R}$ defined by $f(x) = 1/x$ is continuous in its domain, but not uniformly.

A more general definition of uniform continuity applies to functions between metric spaces (there are even more general environments for uniformly continuous functions, i.e. uniform spaces). Given a function $f: X \rightarrow Y$ , where $X$ and $Y$ are metric spaces with distances $d_X$ and $d_Y$ , we say that $f$ is uniformly continuous if$$ \forall \varepsilon > 0\ \exists \delta > 0\ \forall x,y \in X\ d_X(x,y) < \delta \Rightarrow d_Y(f(x),f(y)) < \varepsilon.$$

Uniformly continuous functions have the property that they map Cauchy sequences to Cauchy sequences and that they preserve uniform convergence of sequences of functions.

Any continuous function defined on a compact space is uniformly continuous (see Heine-Cantor theorem).