$Problem:$
Let $G_1$, $G_2$ be regions in $\mathbb{C}$ such that $G_1\cap G_2$ is connected. Suppose $f : G_1\cup G_2 \to \mathbb{C}$ is continuous and $\int_{\gamma} f(z)dz=0$ for every closed path $\gamma$ in $G_1$ and for every closed path $\gamma$ in $G_2$. Show that then this equality holds for every closed path $\gamma$ in $G_1\cup G_2$ as well.
Where a region is an open connected in $\mathbb{C}$
If the curve is contained in $ G_1 $ or $ G_2 $ there is nothing to prove. In the most important case what can I do, thanks for reading, all suggestions or help are welcome.