real analysis - Differentiability and Continuity on an open interval

Let $f:[0,\infty)\to\mathbb{R}$ be defined by:

$$\begin{cases}&x\sin(\frac{1}{x}) \, \, &\text{if}\, \, x > 0\\ & 0, &\text{if} \, \, x = 0\end{cases}$$

Show that $f$ is continuous on $[0,\infty)$ and differentiable on $(0,\infty)$. Also, show that $f$ has no local maximum or minimum in the endpoint $x = 0$ of the domain of $f$.

I can manage to prove continuity on a single point using the epsilon-delta technique, although the intervals here were a surprise, how do you go about proving such a thing? And any hints about the second part of the problem would also be appreciated.