Holomorphic Function is Continuously Differentiable

Theorem
Let $D \subseteq \C$ be an open set.

Let $f : D \to \C$ be a holomorphic function.

Then $f$ is continuously differentiable in $D$.

Proof
Cauchy's Integral Formula shows that the derivatives $f'$ and $f''$ exist.

It follows that $f'$ is complex-differentiable.

Complex-Differentiable Function is Continuous shows that $f'$ is continuous.