(i) State the Mean Value Theorem. Use it to show that if $f:(a, b) \rightarrow \mathbb{R}$ is a differentiable function whose derivative is identically zero, then $f$ is constant.

(ii) Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function and $\alpha>0$ a real number such that for all $x, y \in \mathbb{R}$,

$$|f(x)-f(y)| \leqslant|x-y|^{\alpha} .$$

Show that $f$ is continuous. Show moreover that if $\alpha>1$ then $f$ is constant.

(iii) Let $f:[a, b] \rightarrow \mathbb{R}$ be continuous, and differentiable on $(a, b)$. Assume also that the right derivative of $f$ at $a$ exists; that is, the limit

$$\lim _{x \rightarrow a+} \frac{f(x)-f(a)}{x-a}$$

exists. Show that for any $\epsilon>0$ there exists $x \in(a, b)$ satisfying

$$\left|\frac{f(x)-f(a)}{x-a}-f^{\prime}(x)\right|<\epsilon .$$

[You should not assume that $f^{\prime}$ is continuous.]