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Let $D \subset \mathbb{C}$ be a domain and let $a \in D$. Suppose $f: D \smallsetminus \{a\} \to \mathbb{C}$ is analytic and that $a$ is an essential singularity of $f$. Show that $f$ cannot be univalent (= injective) in any neighborhood of $a$.

This is a trivial consequence of the Picard theorem. But I don't know if there is any elementary approach.

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    Please edit your post so that the "title" is a succinct description of your question, and the body of your question is contained entirely in the body text. Don't split the question text partly between the title and the main part. And please punctuate correctly. Also, if you enclose mathematics expressions in dollar signs `$`, you can make use of our [built-in LaTeX mathematics display]( http://meta.math.stackexchange.com/questions/1773/do-we-have-an-equation-editing-howto).2011-07-23
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    [Casorati-Weierstrass](http://en.wikipedia.org/wiki/Casorati%2DWeierstrass_theorem)2011-07-23
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    By the way: proving this via Picard is cracking a nut with a sledgehammer. Now that you know how the result is called you should be able to locate it in *any* book on complex analysis.2011-07-23
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    Ah, looks like @Theo cleaned up the question for you.2011-07-23
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    The case where the essential singularity is at infinity was covered in answers to two previous questions: http://math.stackexchange.com/questions/39479/entire-bijection-of-mathbbc-with-2-fixed-points/39487#39487 and http://math.stackexchange.com/questions/29758/entire-1-1-function/29762#29762. (In each of those it was assumed that the function is entire, but the idea is the same as in Theo's answer here.)2011-07-28

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