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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?

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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.


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 $\{1/n\}\subset \Bbb R$.

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  • $\begingroup$ @ArpitKansal Sorry that I made a mistake in my previous comment. In general there doesn't. $\endgroup$ – Vim Aug 23 '15 at 7:18
  • $\begingroup$ @Vim,Thanks a lot! $\endgroup$ – Arpit Kansal Aug 23 '15 at 8:07
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    $\begingroup$ @ArpitKansal You're very welcome $\endgroup$ – Vim Aug 23 '15 at 8:08
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$f(x)=\frac{1}{x}$ is continuous on $(0,1)$ but not uniformly continuous.

Proof: The proof is by $\epsilon, \delta$ method, by choosing $\delta= \frac{1}{x_0}$ we get that the $\delta$ depend to the point of continuity so the continuity is not uniform.

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