Differentiability implies continuity

Differentiability implies continuity: Let $f:I\to\mathbb{R}$ be a function on an interval $I$, and let $a\in I$ be an interior point. If $f$ is differentiable at $a$, then

so $f$ is continuous at $a$.

More generally, if $f:U\to\mathbb{R}^m$ is differentiable at $a\in U$ in the sense of the Fréchet derivative, then $f$ is continuous at $a$ (i.e. a continuous map at that point). This connects derivatives to limits at a point.