Chapter 2.8, Problem 64E

Left- and Right-Hand Derivatives The left-hand and right-hand derivatives of $f$ at $a$ are defined by $f_{-}^{\text{'}}(a)=\underset{h\to 0^{-}}{\lim }\frac{f(a+h)-f(a)}{h}$ and $f^{\prime }(a)=\underset{h\to 0^{+}}{\lim }\frac{f(a+h)-f(a)}{h}$

if these limits exist. Then $f\prime (a)$ exists if and only if these onesided derivatives exist and are equal.

64. Find $f\prime -(0)$ and $f_{+}\prime (0)$ for the given function $f$ . Is $f$ differentiable at 0?

(a) $f(x)=\left\{\begin{array}{ll}0\hfill & \text{ if }x⩽0\hfill \\ x\hfill & \text{ if }x>0\hfill \end{array}\right.$

(b) $f(x)=\left\{\begin{array}{ll}0\hfill & \text{ if }x⩽0\hfill \\ x^2\hfill & \text{ if }x>0\hfill \end{array}\right.$