Assuming that $\sin x$ is continuous on $\mathbb R$, find all real $\alpha$ such that $x^\alpha\sin (1/x)$ is uniformly continuous on the open interval (0,1).
I'm guessing that I need to show that $x^\alpha\sin x$ is continuously extendable to [0,1]. Doing that for $x=1$ is pretty trivial, but I am having trouble doing that for $x=0$. I believe that the $\lim_{x\to 0}x^\alpha\sin (1/x)=0$, but how can I find what $f(0)$ equals?
I would appreciate any guidance! Thanks for your help in advance.