Consider the following function:
$$f(x) = \begin{cases} x^{2} \cos\left(\frac{1}{x}\right), & x \neq 0 \\ a, & x = 0 \end{cases}$$
(a) Find the value of $a$ for which $f$ is continuous.\\
Use this value of $a$ to calculate the following.\\
(b) $f'(0)$.\\
(c) $\lim_{x \rightarrow 0} f'(x)$.