Differentiable Implies Continuity

Theorem

1. if $f(x)$ is Differentiable
2. then, $f(x)$ is Continuous

Open Set Version

Suppose that $f:U\subset {\mathbb{R}}^n\to {\mathbb{R}}^k$ is differentiable at $x_0\in U$ where $U$ is an Open Set. Then, $f$ is continuous at $x_0$.

Proof

1. Assume $f$ is differentiable
2. Then $f^{\prime }(x)=$