Let $D$ be a domain in $\mathbb C$ that is symmetric about $ \mathbb{R}$. Let $$D_{-}=\{ z \in D: \operatorname{Im} z<0\},\ D_+=\{ z \in D: \operatorname{Im} z>0\}.$$
Assume $f $ is continuous function on $D_+ \cup I$, with $I=D \cap \mathbb{R}$, and $f$ is analytic. Define $F$ on all $D$ by$$F(z)=f(z) $$for $z \in D_+ \cup I $, and $$F(z)=\overline{f(\overline z)}$$ for $z \in D$.
I want to show if $f $ is a real-valued function on $I$ then $f$ is analytic in $D$.
I showed $F$ is continuous. Then I am using Morera's theorem by considering some rectangles $R$ to show that $F$ is uniformly continuous on $\mathbb{R}$, but I am really getting stuck with that and with the rest of the proof.
So I would appreciate any help with that.
I suggest you refer to Schwarz Reflection Principle. It is essentially the same statement.