(a) Let $n \geqslant 1$ and $f$ be a function $\mathbb{R} \rightarrow \mathbb{R}$. Define carefully what it means for $f$ to be $n$ times differentiable at a point $x_{0} \in \mathbb{R}$.

$$\text { Set } \operatorname{sign}(x)= \begin{cases}x /|x|, & x \neq 0 \\ 0, & x=0 .\end{cases}$$

Consider the function $f(x)$ on the real line, with $f(0)=0$ and

$$f(x)=x^{2} \operatorname{sign}(x)\left|\cos \frac{\pi}{x}\right|, \quad x \neq 0 .$$

(b) Is $f(x)$ differentiable at $x=0$ ?

(c) Show that $f(x)$ has points of non-differentiability in any neighbourhood of $x=0$.

(d) Prove that, in any finite interval $I$, the derivative $f^{\prime}(x)$, at the points $x \in I$ where it exists, is bounded: $\left|f^{\prime}(x)\right| \leqslant C$ where $C$ depends on $I$.