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.

Proof sketch: Differentiability at $a$ means there is a linear map $A:\mathbb{R}^n\to\mathbb{R}^m$ such that $\lim_{h\to 0}\frac{\|f(a+h)-f(a)-Ah\|}{\|h\|}=0.$ Hence $f(a+h)-f(a)=Ah+o(\|h\|)$. Taking norms and letting $h\to 0$ gives $\|f(a+h)-f(a)\|\to 0$, which is continuity at $a$.