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

f ' ( a ) = lim h → 0 − f ( a + h ) − f ( a ) a   and   f ' + ( a ) = lim h → 0 + 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 ' − ( 4 ) and f ' + ( 4 ) for the function

f ( x ) = { 0     if   x ≤ 0 5 − x   if   0 < x < 4 1 5 − x   if   x ≥ 4

(b) Sketch the graph of f

(c) Where is f discontinuous?

(d) Where is f not differentiable?