Is a bounded and continuous function uniformly continuous?

Question

$f\colon(-1,1)\rightarrow \mathbb{R}$ is bounded and continuous does it mean that $f$ is uniformly continuous?

Well, $f(x)=x\sin(1/x)$ does the job for counterexample? Please help!

Answer 1

You're close: $$\sin\frac{1}{x+1}$$ is a counterexample to the statement.

Answer 2

For continuity to lead to uniform continuity, domain has to be compact, and as you can see the domain is not compact here. Also, rightly $f(x)=\sin(\frac{1}{x+1}) $ serves as a counterexample.