Define what it means for a function of a real variable to be differentiable at $x \in \mathbb{R}$.

Prove that if a function is differentiable at $x \in \mathbb{R}$, then it is continuous there.

Show directly from the definition that the function

$$f(x)= \begin{cases}x^{2} \sin (1 / x) & x \neq 0 \\ 0 & x=0\end{cases}$$

is differentiable at 0 with derivative 0 .

Show that the derivative $f^{\prime}(x)$ is not continuous at 0 .