# Definition:Uniform Continuity/Metric Space

## Definition

Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces.

Then a mapping $f: A_1 \to A_2$ is uniformly continuous on $A_1$ if and only if:

$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x, y \in A_1: \map {d_1} {x, y} < \delta \implies \map {d_2} {\map f x, \map f y} < \epsilon$

where $\R_{>0}$ denotes the set of all strictly positive real numbers.

## Also see

• Results about uniform continuity can be found here.