Holomorphic Function is Continuously Differentiable

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

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

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

Proof
Let $z \in U$.

The derivatives $\map {f'} z$ and $\map {f''} z$ exist by Cauchy's Integral Formula for Derivatives.

Therefore $f'$ is continuous at $z$ by Complex-Differentiable Function is Continuous.