TITLE: Verification: Application of the Identity Theorem $f(\mathbb{R}) = \mathbb{R} \implies f(z) = \overline{f(\bar z)}$
QUESTION [1 upvotes]: I want to prove that if $f(\mathbb{R}) = \mathbb{R}$ then $f(z) = \overline{f(\bar z)}$ for an entire function $f$.
My idea was using the identity theorem. If $f(\mathbb{R}) = \mathbb{R}$ then:
$$\forall z \in \mathbb{R}: f(z) = f(\bar z) = \overline{f (\bar z)}$$
I only just learned about the identity theorem, so I wanted to ask whether this is indeed a valid application of it.

REPLY [4 votes]: Yes, it is, after you have proved that the function $z\mapsto\overline{f\left(\overline z\right)}$ is an analytic function.
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