Definition:Uniform Continuity/Metric Space

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Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be metric spaces.

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

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

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

Also see

  • Results about uniform continuity can be found here.