Proving if a function is continuous or not Can somebody explain how to solve this problem:
The first question is asking you to show that
$$\lim_{x \to 0} x \sin \frac{1}{x} = 0.$$

Hint:


   Use the fact that $-1 \le \sin \frac{1}{x} \le 1$ for all $x \ne 0$.




For the second question, you just need to determine whether
$$\lim_{x \to 0} \sin \frac{1}{x}$$
is equal to $0$ or not.
Let $\epsilon &gt;0$
Clearly $|f(x)-f(0)|&lt;x$ for all $x$
if we set $\delta=\epsilon$ we obtain 
$|x-0|=|x|&lt; \delta \Rightarrow |f(x)-f(0)|&lt;x$