Entire function. Prove that $f(\bar{z})=\overline{f(z)}, \forall z\in C$

Question

Let $f$ a entire function: $f(R)\subset R.\;$ Prove that $f(\bar{z})=\overline{f(z)}, \forall z\in C$

Answer 1

The function $g(z):=\overline{f(\bar{z})}$ is also entire. By assumption, it coincides with $f$ on $\mathbb{R}$. By the principle of isolated zeroes, it follows that $f=g$ on $\mathbb{C}$.