Given any non compact set $A \subset \mathbb R^n$ does there exist a continuous function $f: A \to \mathbb R$ which is not uniformly continuous?

Question

It's well known that if $ A \subset \mathbb R^n$ is compact then every continuous function $f:A \to \mathbb R$ is uniformly continuous.So the obvious question is:

Given a non compact set $A \subset \mathbb R^n$ does there always exist a continuous function $f: A \to \mathbb R$ which is not uniformly continuous?

Any ideas?

Answer 1

There doesn't. Consider the case where $A=\Bbb Z\subset \Bbb R$, then any function $f$ defined on $\Bbb Z$ is uniformly continuous.

In general, any function defined on a susbset of $\Bbb R^n$ consisting solely of isolated points is uniformly continuous.

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EDIT: CORRECTION
Today I just accidentally came across this one year old answer I posted and I spotted a fatal flaw in the second paragraph: In fact, even if the domain consists solely of isolated points in a metric space, the function defined on it need not be uniformly continuous at all (but continuity is still guaranteed). A very quick example is $f(x)=1/x$ defined on $\{1/n\}\subset \Bbb R$.