# Paper 1, Section II, $10 F$

(a) Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be differentiable at $x_{0} \in \mathbb{R}$. Show that $f$ is continuous at $x_{0}$.

(b) State the Mean Value Theorem. Prove the following inequalities:

$\left|\cos \left(e^{-x}\right)-\cos \left(e^{-y}\right)\right| \leqslant|x-y| \quad \text { for } x, y \geqslant 0$

and

$\log (1+x) \leqslant \frac{x}{\sqrt{1+x}} \text { for } x \geqslant 0 .$

(c) Determine at which points the following functions from $\mathbb{R}$ to $\mathbb{R}$ are differentiable, and find their derivatives at the points at which they are differentiable:

$f(x)=\left\{\begin{array}{ll} |x|^{x} & \text { if } x \neq 0 \\ 1 & \text { if } x=0, \end{array} \quad g(x)=\cos (|x|), \quad h(x)=x|x|\right.$

(d) Determine the points at which the following function from $\mathbb{R}$ to $\mathbb{R}$ is continuous:

$f(x)= \begin{cases}0 & \text { if } x \notin \mathbb{Q} \text { or } x=0 \\ 1 / q & \text { if } x=p / q \text { where } p \in \mathbb{Z} \backslash\{0\} \text { and } q \in \mathbb{N} \text { are relatively prime. }\end{cases}$