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

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?

• What are the open balls of radius $1/2$ in a discrete space? – Prahlad Vaidyanathan Oct 1 '13 at 9:47
• 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. – tomasz Jul 23 '17 at 11:51

Hint: For any $\varepsilon>0$ put $\delta:=\dfrac12$ in the definition of uniform continuity.
• If $d(x,y)<\frac12$ then $x=y$, hence $|f(x)-f(y)|=0<\varepsilon$. – njguliyev Oct 1 '13 at 10:08