Differentiability and Continuity

One-way implication

Differentiable $\Rightarrow$ continuous. The converse is false. A function can be perfectly continuous yet fail to have a derivative, and AP tests exactly the three ways this happens:

• Corner — the one-sided derivatives exist but differ, e.g. $|x|$ at $0$.
• Cusp — one side heads to $+\infty$, the other to $-\infty$, e.g. $x^{2/3}$ at $0$.
• Vertical tangent — both sides head to $+\infty$ (or both $-\infty$), e.g. $x^{1/3}$ at $0$.

Contrapositive form, worth memorising because free-response uses it: if $f$ is not continuous at $a$, then $f$ is not differentiable at $a$. That single line closes many justification questions.