Hi,

I'm a little stuck with this exercise.

Let $\displaystyle f:\mathbb{C}\to \mathbb{C}$ be a holomorphic function which satisfies $\displaystyle f(z)\in\mathbb{R}\quad\forall z\in\mathbb{R}$. I want to show that

$\displaystyle f(\overline{z}) = \overline{f(z)}$.

We do have a non-discrete set $\displaystyle \mathbb{R}$ on which this identity holds, so if both functions were holomorphic, the proof was complete using the identity theorem.

But why are the functions $\displaystyle z\mapsto f(\overline{z})$ and $\displaystyle z\mapsto \overline{f(z)}$ holomorphic?

Thank you!