Application of Morera's theorem to constructed function

Question

Say we have a function $f(z)$ which is holomorphic on a domain $G\subseteq\mathbb{C}$. Let $a\in G$. If we define \begin{equation}g(z)=\begin{cases}\frac{f(z)-f(a)}{z-a}&&\text{if}&&z\neq a\\f'(z)&&\text{if}&&z=a\end{cases} \end{equation} clearly $g(z)$ is holomorphic everywhere on $G$ except maybe at $a$. It was claimed in my notes that Morera's theorem can be used here to show functions 'of this type' are differentiable at $a$, but I can't seem to find the reasoning as to why. How is Morera's theorem applied here?

Answer 1

Clearly $g$ is continuous on $G$. Let $T$ be a triangle contained in $G$ and the $\partial T$ be its boundary. Then $\int_{\partial T}g(z)\,dz=0$. To prove this you will have to consider the position of $a$ with respect to $T$.

• If $a$ is outside $T$, then $\int_{\partial T}g(z)\,dz=0$ by Cauchy's theorem.
• If $a$ is a vertex of $T$, given $\epsilon>0$ divide $T$ into triangles, one of which has $T$ as a vertex and area $\le\epsilon$. Then $\bigl|\int_{\partial T}g(z)\,dz\bigr|\le C\,\epsilon$ for some constant $C$ depending only on $g$.
• Finally, if $a$ is in the interior of $T$, divide $T$ into triangles with one vertex at $a$.