This is from Conway's book:
Suppse $f:G \rightarrow \mathbb{C}$ is analytic and define $\phi: G \times G \rightarrow \mathbb{C}$ by $\phi(z,w)=\frac{f(z)-f(w)}{(z-w)}$ if $z \neq w$ and \phi(z,z)=f'(z). Prove that $\phi$ is continuous...
Because $f$ is analytic, f' exists for all points $z$ in the region $G \times G$.
I think I get for free that f' is continuous. But I'm not sure where to go from here.