Could any one tell me how to solve this one?
$U$ be open connected subset of $\mathbb{C}$ such that $z\in U\Rightarrow\bar{z}\in U$. Let $f$ be holomorphic function on $U$, if $f$ is real valued on a non de-generate sub interval of $U\cap\mathbb{R}$, then we need to show $f$ is real valued on $U\cap\mathbb{R}$
I got this question as a past years quals, I dont kow what is non de generate sub interval.
With problems that involve reflection (such as $r(z)= \bar z$), it helps to remember how to reflect holomorphic functions: $r\circ f\circ r$ is the reflection of $f$. Here, the reflection of $f$ is $\overline{f(\bar z)}$, which is holomorphic in $U$ (check this). If you are still stuck, see below.
Since $f(z)=\overline{f(\bar z)}$ on a set with a limit point in $U$, it follows (by the identity theorem) that $f(z)=\overline{f(\bar z)}$ for all $z\in U$. The statement about $z\in U\cap \mathbb R$ follows at once.
