vertical at x = c; there cannot be a corner or cusp at x = c.
Each of the “prohibitions” in the preceding paragraph (each “cannot”) tells
how a function may fail to have a derivative at c. These cases are illustrated in
Figures N3–2 (a) through (f).
Figure N3–2The graph in (e) is for the absolute function, f (x) = |x|. Since f ′(x) = −1 for all
negative x but f ′(x) = +1 for all positive x, f ′(0) does not exist.
We may conclude from the preceding discussion that, although
differentiability implies continuity, the converse is false. The functions in (d),
(e), and (f) in Figure N3–2 are all continuous at x = 0, but not one of them is