AP Calculus AB and BC glossary

Differentiability

Also called: Differentiable

A function is differentiable at a point when the limit defining the derivative exists there, which requires the graph to be locally smooth with a single well-defined tangent slope. Differentiability always implies continuity, but continuity does not imply differentiability.

Differentiability fails in four recognizable ways: a corner, where the one-sided slopes disagree; a cusp, where they run off to opposite infinities; a vertical tangent, where the slope is infinite; and any discontinuity, which rules out a derivative immediately.

The implication runs one way only. Because $|x|$ is continuous at 0 but has slopes of $-1$ and $1$ on either side, it is a permanent counterexample to the converse.

For piecewise functions

Making a piecewise function differentiable at a seam takes two equations, not one: the pieces must meet, and their derivatives must match there.

Appears in: Unit 2: Defining the Derivative