real analysis - Continuity and Differentiation on open interval

$$f(x) = \begin{cases} x\sin(1/x), & \text{if $x$ $\ne$ $0$} \\ 0, & \text{if $x$ = $0$} \\ \end{cases}$$

Is $f$ continuous on $(-1/\pi$, 1/$\pi$)?
Is $f$ differentiable on $(-1/\pi$, 1/$\pi$)?

I know how to prove continuity on a single point, but I'm not sure how to prove continuity for a whole interval. Also, I know there is a theorem that states that if a function is differentiable at a point, then it's continuous.