Differentiability implies continuity

Differentiability implies continuity: Let $U\subseteq\mathbb{R}^n$ be open and let $f:U\to\mathbb{R}^m$ be differentiable at $a\in U$. Then $f$ is continuous at $a$.

This is a basic but essential fact: differentiability is a stronger local property than continuity, and many arguments implicitly use it.