AP Calculus AB and BC glossary

Differentiability implies continuity

If a function is differentiable at a point then it is continuous there. The converse is false: a function can be continuous at a point and still fail to be differentiable, which is what happens at a corner, a cusp, or a vertical tangent.

f \text{ differentiable at } a \implies f \text{ continuous at } a

The absolute value function at $x = 0$ is the standard counterexample to the converse. It is continuous there, but the one-sided slopes are $-1$ and $1$ , so no single tangent slope exists.

The mistake

Reading the implication backwards. Continuity does not give you differentiability, and an argument that assumes it will not earn justification credit.

Appears in: Unit 2: Defining the Derivative