$f$ entire and non-constant function, then for fix $x \in \mathbb{C}$ the set of $(z \in \mathbb{C} : f(z)=x)$ is finite or countable.

Question

If $f$ entire and non-constant function, then for fix $x \in \mathbb{C}$ the set of $(z \in \mathbb{C} : f(z)=x)$ is finite or countable.

I was trying to prove this statement by contradiction or using the continuity of $f$ but unsuccessfully.

Answer 1

If it was uncountable, then it would admit an accumulation point. However, then $f$ was constant by the identity theorem.