Let $g: \mathbb{C} \rightarrow \mathbb{C}$ be a continuous function such that

$$\int_{\Gamma} g(z) d z=0$$

for any closed curve $\Gamma$ which is the boundary of a rectangle in $\mathbb{C}$ with sides parallel to the real and imaginary axes. Prove that $g$ is analytic.

Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be continuous. Suppose in addition that $f$ is analytic at every point $z \in \mathbb{C}$ with non-zero imaginary part. Show that $f$ is analytic at every point in $\mathbb{C} .$

Let $\mathbb{H}$ be the upper half-plane of complex numbers $z$ with positive imaginary part $\Im(z)>0$. Consider a continuous function $F: \mathbb{H} \cup \mathbb{R} \rightarrow \mathbb{C}$ such that $F$ is analytic on $\mathbb{H}$ and $F(\mathbb{R}) \subset \mathbb{R}$. Define $f: \mathbb{C} \rightarrow \mathbb{C}$ by

$$f(z)= \begin{cases}F(z) & \text { if } \Im(z) \geqslant 0 \\ \overline{F(\bar{z})} & \text { if } \Im(z) \leqslant 0\end{cases}$$

Show that $f$ is analytic.