TITLE: Is a bounded and continuous function uniformly continuous?
QUESTION [24 upvotes]: $f\colon(-1,1)\rightarrow \mathbb{R}$ 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!

REPLY [4 votes]: $\sin(x^2)$ is also a nice example and it's happening because it's not periodic.