For each of the following two functions $f: \mathbb{R} \rightarrow \mathbb{R}$, determine the set of points at which $f$ is continuous, and also the set of points at which $f$ is differentiable.

$$\begin{aligned} &\text { (i) } f(x)= \begin{cases}x & \text { if } x \in \mathbb{Q} \\ -x & \text { if } x \notin \mathbb{Q}\end{cases} \\ &\text { (ii) } f(x)= \begin{cases}x \sin (1 / x) & \text { if } x \neq 0 \\ 0 & \text { if } x=0\end{cases} \end{aligned}$$

By modifying the function in (i), or otherwise, find a function (not necessarily continuous) $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $f$ is differentiable at 0 and nowhere else.

Find a continuous function $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $f$ is not differentiable at the points $1 / 2,1 / 3,1 / 4, \ldots$, but is differentiable at all other points.