This was from Freitag's p247.
Let $D \subseteq \mathbb{C}$, $f:D \rightarrow \mathbb{C}$ be analytic (holomorphic) function. $\phi:D \times D \rightarrow \mathbb{C}$ defined by $ \phi (\zeta,z) = \frac{f(\zeta)-f(z)}{\zeta - z}$ if $\zeta \not= z$ and $\phi(z,z) = f'(z)$ if $\zeta = z$. We have shown that this function is analytic.
Lemma B.3(p247): $G: D \rightarrow \mathbb{C}$, $G(z) = \int_\alpha \phi(\zeta, z) \, d\zeta$ is analytic in $D$ by the Leibniz rule.
My problem is the second statement about analyticity of $G$. For us to apply Leibniz Rule, we need $\alpha$ to be smooth (?) so we can write the integral as $\int_a^b \phi(\alpha(t),z) \alpha'(t) \, dt $. But $\alpha$ is an arbitrary closed curve here.