$\displaystyle
f(x) = \left\{
\begin{array}{lr}
x^2 sin\frac{1}{x} & : x \neq 0\\
0 & : x = 0
\end{array}
\right.
$
I don't think your professor expects you to do all that, but if he does, you need to crank out the $\displaystyle \epsilon$-$\displaystyle \delta$ proofs for each one.
For $\displaystyle x^2$ let $\displaystyle \delta=\min\left\{1,\frac{1}{1+2|x_0|}\right\}$, because $\displaystyle |x-x_0|<1\implies |x+x_0|<1+2|x_0|\implies |x-x_0||x+x_0|<\delta(1+2|x_0|)$
For $\displaystyle \sin x$, you can probably Google a proof. The idea is to let $\displaystyle \delta=\epsilon$ and then use a difference-to-product identity to convert $\displaystyle |\sin x-\sin x_0|$ into something more manageable.
$\displaystyle \frac{1}{x}$ is kind of a pain in the ass, and I've screwed up continuity proofs of this function before. There are probably other threads on MHF with proofs of this.