Chapter 3.2, Problem 31E

One-sided derivatives The right-sided and left-sided derivatives of a function at a point a are given by

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

respectively, provided these limits exist. The derivative f′(a) exists if and only if f₊′(a) = f₋′(a).

1. a. Sketch the following functions.
2. b. Compute f₊′(a) and f₋′(a) at the given point a.
3. c. Is f continuous at a? Is f differentiable at a?

31. f(x) = |x − 2|; a = 2