If a function is continuous on $\mathbb R,$ does it follow that it is uniformly continuous on $(-1,1)?$

Question

I've been trying to think of counterexamples but the ideas I've had so far like $1/(x+1)$ and $\sin(1/(x+1))$ don't work because those aren't continuous on all of $\mathbb R.$

Answer 1

Since it is continuous on the compact set $[-1,1]$, it is uniformly continuous on $[-1,1]$ and hence uniformly continuous on $(-1,1)$.

You can actually prove the converse. If $f$ is uniformly continuous on $(-1,1)$, then it can be extended to be uniformly continuous on $[-1,1].$