When proving Cauchy's integral formula we define:
$$ G(z)= \begin{cases} \frac{f(z)-f(z_0)}{z-z_0}, \text{ } z\neq z_0\\ f'(z_0), \text{ } z=z_0 \end{cases} $$
To continue we have to show that $G(z)$ is analytic except to $z_0$.
So in the first case it is as a quotient of to analytic functions. and in the second it is continuous as it is differentiable at $z_0$
So we overall showed that $G(z)$ is continuous everywhere in $D$ and nalytic except to $z_0$?
If $D$ is the (open) domain of $f$ and if $z_0 \in D$, then $G$ is analytic on $D \setminus \{z_0\}$ and continuous on $D$.