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Let $f$ be a nonconstant entire function and $U$ be an open set in the plane. Show that there is a $z_0$ such that $f\left(z_0\right)\in U$.

This question is an exercise for the Maximum Modulus and Mean Value section. I can't figure out how to prove this. I'm more than sure this requires an application of the mean value theorem, but I don't exactly know how to use it.

Any suggestions/tips on how to proceed?

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  • $\begingroup$ Note that this does not imply that $f$ is actually surjective, only that it has dense image. $\exp$ for instance is not surjective, but its image is the punctured plane, which is dense. $\endgroup$ – Ian Dec 9 '14 at 0:21
  • $\begingroup$ Oh wow I didn't see that! I see now why I originally thought that and why I was wrong. $\endgroup$ – Arturo don Juan Dec 9 '14 at 0:33
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Hint: consider $1/(f(z) - u)$, and use Liouville's Theorem.

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  • $\begingroup$ Could you explain what $u$ is? Thanks for the response by the way. $\endgroup$ – Arturo don Juan Dec 9 '14 at 0:16
  • $\begingroup$ Suppose $f(z) \notin U$ for all $z$. Pick $u \in U$, then $f(z)$ is bounded away from $u$... $\endgroup$ – copper.hat Dec 9 '14 at 0:20

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