I'm trying to solve this problem: I have an entire function $f$ which is not a polynomial. I have to prove that there exists a dense subset $\Omega \subset \mathbb{C}$ such that for every $\omega \in \Omega$ the equation $f(z)=w$ has infinitely many solutions.
If $f$ is not a polynomial it must have an essential singularity at infinity, so for every $R$, $f(|z|>R)$ is an open dense subset of the complex plane, but I don't know if this is useful (it's the only thing about 'density' I could think of!). Thanks for any help.