Differentiability implies Continuity

Theorem: Differentiability implies Continuity

Let $f:\mathcal{D}\to {\mathbb{R}}^n$ be a real vector function on an open subset $\mathcal{D}\subseteq {\mathbb{R}}^m$.

If $f$ is differentiable at ${\mathbf{x}}_0\in \mathcal{D}$, then $f$ is also continuous there.

PROOF

TODO