There are three situations where a derivative fails to exist. The derivative of a function at a given point is the slope of the tangent line at that point. So, if you can’t draw a tangent line, there’s no derivative — that happens in cases 1 and 2 below. In case 3, there’s a tangent line, but its slope and the derivative are undefined.

The three situations are shown in the following list.

1. 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.

2. When there’s no tangent line and thus no derivative at a sharp corner on a function. See function f in the above figure.

3. 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.