Problem 62E

62. The left-hand and right-hand derivatives of $f$ at $a$ are defined by and

$$\begin{array}{l}f^{\prime}-(a)=\lim _{k \rightarrow 0^{-}} \frac{f(a+h)-f(a)}{h} \\f^{\prime}+(a)=\lim _{k \rightarrow 0^{+}} \frac{f(a+h)-f(a)}{h}\end{array}$$

if these limits exist. Then $f^{\prime}(a)$ exists if and only if these one-sided derivatives exist and are equal.

(a) Find $f^{\prime}-(4)$ and $f_{+}^{\prime}(4)$ for the function

$$f(x)=\left\{\begin{array}{ll}0 & \text { if } x \leqslant 0 \\5-x & \text { if } 0<x<4 \\\frac{1}{5-x} & \text { if } x \geqslant 4\end{array}\right.$$

(b) Sketch the graph of $f$.

(c) Where is $f$ discontinuous?

(d) Where is $f$ not differentiable?

Step-by-Step Solution