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Let $f: U\to f(U)\subset \Bbb C$ be a holomorphic and nonconstant ( thus in particular an open map), and such that it can be extended continuously on $\overline{U}$.

Where $U$ is a bounded domain of $\Bbb C$ i.e an open and connectedness bounded set. Well I want to know if it's true that the boundary of $U$ is mapped onto ( surjective) the boundary $ f(U)$.

Well at least we know that the image of the boundary of $U$, is contained in the boundary of $f(U)$, i.e $f(\partial(U)) \subset \partial (f(U))$. And that is clear from the fact that $f$ is an open map. (I only used that). I want to know the other containment.

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    $f(\partial U)\subset \partial f(U)$ is incorrect. Can you find some counter-example?2012-11-06

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