Let $V\subset \mathbb{C}$ be open and $f\colon V\to \mathbb{C}$ a continuous function, and assume
$$ \int_{\gamma} f(z) \, dz =0 $$
for any closed contour $\gamma \in V.$ We need to show that $f$ has an antiderivative $F$ defined on $V$.
I know how to prove it when $V$ is a convex open set and $\gamma \in V$ is a boundary of any triangle in $V.$ I do not know how to connect these ideas. Could you help me with this problem?