Textbook Problem

The left-hand and right-hand derivatives off at a are defined by

$f\text{'}(a)=\underset{h\to 0^{-}}{\lim }\frac{f(a+h)-f(a)}{a}\text{and}f{\text{'}}_{+}(a)=\underset{h\to 0^{+}}{\lim }\frac{f(a+h)-f(a)}{h}$

if these limits exist. Then f'(a) exists if and only if these one-sided derivatives exist and are equal.

(a) Find $f^{\text{'}}{}_{-}(4)$ and $f^{\text{'}}{}_{+}(4)$ for the function

$f(x)=\{\begin{array}{l}0\text{if}x\le 0\\ 5-x\text{if}0<x<4\\ \frac{1}{5-x}\text{if}x\ge 4\end{array}$

(b) Sketch the graph of f

(c) Where is f discontinuous?

(d) Where is f not differentiable?