Let $x_{0} \in \mathbb{R}$ and let $f: \mathbb{R} ightarrow \mathbb{R}$ be a function for which the set $\left\{\frac{f(x)-f\left(x_{0} ight)}{x-x_{0}}: x eq x_{0} ight\}$ is bounded. (a) Give an example of a function satisfying the above condition for some $x_{0} \in \mathbb{R}$. (b) For every $\delta>0$, consider the set $$ A_{\delta}=\left\{\frac{f(x)-f\left(x_{0} ight)}{x-x_{0}}: 0<\left|x-x_{0} ight|<\delta ight\} $$ and let $M(\delta)=\sup \left(A_{\delta} ight)$ and $m(\delta)=\inf \left(A_{\delta} ight)$ for every $\delta>0$. Prove that the functions $M, m$ are well-defined, bounded and monotone on $(0, \infty)$, and conclude that the $\operatorname{limits} \lim _{\delta ightarrow 0^{+}} M(\delta)$ and $\lim _{\delta ightarrow 0^{+}} m(\delta)$ exist.

Hint: use question 3a from exercise 2 and question 3 from exercise $8 .$ (c) We denote $\bar{D} f\left(x_{0} ight)=\lim _{\delta ightarrow 0^{+}} M(\delta)$ and $\underline{D} f\left(x_{0} ight)=\lim _{\delta ightarrow 0^{+}} m(\delta)$. Prove that $f$ is differentiable at $x_{0}$ if and only if $\underline{D} f\left(x_{0} ight)=\bar{D} f\left(x_{0} ight)$, and in that case $f^{\prime}\left(x_{0} ight)=\underline{D} f\left(x_{0} ight)=\bar{D} f\left(x_{0} ight)$.