# Image of $f(A)$ when $A$ is compact and $f$ is analytic in the interior of $A$ and continuous on the boundary of $A$

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

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