Suppose there is a non constant complex function $f$ and a compact set $A$ such that $f$ is analytic in the interior of $A$ and continuous on the boundary of $A$. Then clearly $f(A)$ is compact. Now from the maximum modulus principal we know that $|f|$ attains a maximum on A. But my question is can an element in $\mathring{A}$ be mapped onto $f(\overline{A}-A)$. That is, is it possible for a interior point of that compact set be mapped to a boundary point of the image? I do not know whether this is true or false but just feel like this cannot happen for non constant analytic functions. Can someone help to prove or disprove this? Thanks

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    $\begingroup$ Are you familiar with the open mapping theorem? $\endgroup$ – Brandon Carter Nov 10 '14 at 15:47
  • $\begingroup$ @BrandonCarter Yes I am $\endgroup$ – user81883 Nov 10 '14 at 15:49
  • $\begingroup$ @BrandonCarter I see your point now $\endgroup$ – user81883 Nov 10 '14 at 15:50

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