Uniform continuity

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• Don't forget a uniformly continuous map is continuous and co - Alec (talk) 21:14, 8 April 2017 (UTC)

Definition

Let $(X,d_1)$ and $(Y,d_2)$ be metric spaces and let $f:X\rightarrow Y$ be a map between them. We say $f$ is uniformly continuous if:

• $\forall\epsilon>0\exists\delta>0\forall x,y\in X\big[d_1(x,y)<\delta\implies d_2(f(x),f(y))<\epsilon\big]$
  • For comparison: continuity at $x\in X$ (in a map between metric spaces) is $\forall\epsilon>0\exists\delta>0\forall y\in X\big[d_1(x,y)<\delta\implies d_2(f(x),f(y))<\epsilon\big]$ - uniform continuity differs by supposing given an $\epsilon >0$ there is some $\delta>0$ that'll "work" for all $x,y\in X$, not just for a fixed-before-$\epsilon$ $x$.