Differentiability and continuity

Between differentiability and continuity there is a precise implication, but in one direction only: differentiability is stronger than continuity.

Remark — Differentiability and continuity

If $f$ is differentiable at $x_0$, then it is continuous at $x_0$ (a necessary condition). The converse is false: $f(x)=|x|$ is continuous at $0$ but not differentiable (corner point).

Links

Topics: Derivatives
Concepts: Continuity · Derivative · Corner point
Skills: Reasoning by cases