# 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?

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?

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

• @ArpitKansal Sorry that I made a mistake in my previous comment. In general there doesn't. – Vim Aug 23 '15 at 7:18
• @Vim,Thanks a lot! – Arpit Kansal Aug 23 '15 at 8:07
• @ArpitKansal You're very welcome – Vim Aug 23 '15 at 8:08

$f(x)=\frac{1}{x}$ 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.