Preimage of the Real Line

Question

Prove that if $f$ is an analytic function on an open subset $U$ of $\mathbb{C}$ then the non empty preimage of $\mathbb{R}$ under $f$ cannot be compact. I have been trying this problem for a long time now but to no avail. I was trying to prove this by assuming the contrapositive and then showing that $f$ is constant. Can anyone please help me with this? Thanks for any help.

Answer 1

Suppose otherwise. Then $f\bigl(f^{-1}(\mathbb{R})\bigr)$ is a non-empty compact subset of $\mathbb R$. Therefore, there is a $x\in f\bigl(f^{-1}(\mathbb{R})\bigr)$ such that $x=\max f\bigl(f^{-1}(\mathbb{R})\bigr)$. So, $x\in\mathbb R$ but the image of $f$ contains no real number greater than $x$. That's impossible, by the open mapping theorem. Unless of course, $f$ is constant, in which case $f^{-1}(\mathbb{R})=f^{-1}\bigl(\{x\}\bigr)$, which is not compact.