Let $f$ be a continuous function defined on a connected open set $D \subset \mathbb{C}$. Prove carefully that the following statements are equivalent.

(i) There exists a holomorphic function $F$ on $D$ such that $F^{\prime}(z)=f(z)$.

(ii) $\int_{\gamma} f(z) d z=0$ holds for every closed curve $\gamma$ in $D$.