# Application of Maximum Modulus Principle on the Boundary

• May 21st 2009, 11:01 AM
Let U be a connected open set, and $U^c$ its closure. Let f be a continuous function on $U^c$, analytic and nonconstant on U. If $z_0 \in U^c$ is a maximum point for f, then $z_0$ lies on the boundary of $U_c$
It's kind of trivial, if $z_0$ was in $U$ then by the maximum modulus principle $f$ would be constant on $U$ and hence on $U^c$. So unless you were assuming $f$ was constant this is a contradiction, and hence $z_0$ must belong to the boundary of $U$.