Textbook Problem

Let

$f(x)=\{\begin{array}{ll}|x{|}^x\hfill & \text{if}x\ne 0\hfill \\ 1\hfill & \text{if}x=0\hfill \end{array}$

(a) Show that f is continuous at 0.

(b) Investigate graphically whether f is differentiable at 0 by zooming in several limes toward the point (0, 1) on the graph of f.

(c) Show that f is not differentiable at 0. How can you reconcile this fact with the appearance of the graphs in part (b)?