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Proving that every function defined on a discrete metric space is uniformly continuous.

I know that every function on compact set is uniformly continuous, but how about discrete metric space. Can someone please gimme a hand on this please?

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  • $\begingroup$ What are the open balls of radius $1/2$ in a discrete space? $\endgroup$ – Prahlad Vaidyanathan Oct 1 '13 at 9:47
  • $\begingroup$ This is misleading: what it should say is that any function from a discrete metric space to another discrete metric space is uniformly continuous. A function from a discrete metric space to an arbitrary metric space can be very far from uniformly continuous. $\endgroup$ – tomasz Jul 23 '17 at 11:51
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Hint: For any $\varepsilon>0$ put $\delta:=\dfrac12$ in the definition of uniform continuity.

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  • $\begingroup$ Actually can you explain a little bit finding the distance of the ball? $\endgroup$ – user97994 Oct 1 '13 at 10:06
  • $\begingroup$ If $d(x,y)<\frac12$ then $x=y$, hence $|f(x)-f(y)|=0<\varepsilon$. $\endgroup$ – njguliyev Oct 1 '13 at 10:08
  • $\begingroup$ Ohhh duh ok hitting my head with a stick $\endgroup$ – user97994 Oct 1 '13 at 10:10

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