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.

$\blacksquare$

Sources

• 2001: Christian Berg: Kompleks funktionsteori: $\S 4.2$