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Let f be an entire function in the complex plane. Show that if f is constant on a non-empty open set U ⊆ $\mathbb{C}$ then f is constant on $\mathbb{C}$

I think this is a Liouville's theorem question but I don't know how to incorporate it in a formal proof.

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    $\begingroup$ If it more like identity principle. If $f$ is constant on an open then its Taylor series at a point in that open is constant. Finally, Cauchy's theorem tells you that that series converges to the function everywhere. $\endgroup$ – OR. Apr 3 '15 at 1:07
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This has nothing to do with Liouville's theorem. This is called the uniqueness theorem, and it follows from the existence of the power series converging to this function everywhere in the plane.

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