show that for any closed curve $\gamma$ containing $z_0$, $\int _\gamma f=0$

Question

Let $f(z)$ be holomorphic on a simple connected region A, except possibly not holomorphic at $z_0\in A$. Suppose, however, that f is bounded in absolute value near $z_0$. show that for any closed curve $\gamma$ containing $z_0$, $$\int _\gamma f=0$$

please any one help me with this problem i did't have any idea for how to solve this problem please help me

Answer 1

You need the following theorem:

If $f:D \to A$ is holomorphic function, except for $z_{0} \in D$ and f is bounded near $z_{0}$, then f can be extended to holomorphic function for all $z \in D$.

It is known as Riemman's Removable Singularity Theorem.