Is there an example of a complex function on a compact set K that is continuous, non-analytic, and achieves its maximum on int(K)?
Since the max mod principle says that if the function is analytic, then the maximum is on the boundary. I'm looking for a non-analytic function where the maximum is not on the boundary, but the function is continuous.
For any compact set $K \subset \Bbb C$ with nonempty interior and $a \in \operatorname{int}(K)$, the function $f(z) = -|z-a|$ is such an example.
There is nothing special about the complex numbers here, the same example works on any metric space $(X, d)$ with $f(x) = -d(x, a)$ for some $a \in \operatorname{int}(K)$.