The three situations are shown in the following list.
-
When there’s no tangent line and thus no derivative at any of the three types of discontinuity:
-
A removable discontinuity — that’s a fancy term for a hole — like the holes in functions r and s in the above figure.
-
An infinite discontinuity like at x = 3 on function p in the above figure.
-
A jump discontinuity like at x = 3 on function q in the above figure.
Continuity is, therefore, a necessary condition for differentiability. It’s not, however, a sufficient condition as the next two cases show. Dig that logician-speak.
-
-
When there’s no tangent line and thus no derivative at a sharp corner on a function. See function f in the above figure.
-
Where a function has a vertical inflection point. In this case, the slope is undefined and thus the derivative fails to exist. See function g in the above figure.