Is $f$ a polynomial if every preimage of a one point set is bounded? Suppose $f$ is an entire function such that for all $a \in \mathbb{C}$ the set $f^{-1}\{a\}$ is a bounded set,is it true that $f$ is a polynomial?Any hint and idea on this question is appreciated.Thank you.

Hint: Consider $g(z)=f(1/z)$ and apply Casorati–Weierstrass to the singularity at 0.
