Today I have been looking at the Maximum Modulus Principle (MMP). I take it for granted in the following form.
$\quad(*)\quad$ Let $f$ be holomorphic on a domain $\Omega \subset \mathbb{C},$ where "domain" indicates open and connected. Let $\zeta \in \Omega,$ and suppose it has a neighbourhood in which $|f(\zeta)| \geq |f(z)|$ holds. Then $f$ is constant on $\Omega.$
On the wikipedia page for MMP, as of today the 20th of June, there is a corollary of $(*)$ given, and a sketch of the proof. The corollary is called there the "Related statement" (link to it: https://en.wikipedia.org/w/index.php?title=Maximum_modulus_principle&oldid=1009743623#Related_statement ). I tried to write a more detailed proof of how the corollary follows from $(*)$, because I wasn't fully convinced by what I saw there.
My question is whether my proof (below) is actually what the sketch is suggesting. If "yes", then maybe you can verify that my proof is alright. But if "No", what is going on with that sketch? Is there some lemma being implicitly used in the reasoning? I'd like to know what I'm missing.
Proof: Let $D$ be bounded, non-empty, open, so that $\overline{D}$ is compact. We are given that $f$ is continuous on $\overline{D}$ and is holomorphic on $D$. Since $|f|$ is real and continuous on a compact, there is a $z_o \in \overline{D}$ where the modulus attains its maximum. If $z_o \in \partial D$, then we are done; else we have $z_o \in D$. Let $D_m$ be the connected component of $D$ in which $z_o$ lies. (We take it as known that $D$ may be partitioned into a family of disjoint domains.) By $(*)$ we have that $f$ is constant on $D_m$, and it is constant on $\overline{D_m} \subset \overline{D}$, by continuity.
Thus $|f|$ attains its maximum at some $x \in \partial D_m.$ ($\mathbb{C}$ is connected, $D_m$ is bounded; hence $\partial D_m \not = \emptyset.$) Lastly we must show that $x \in \partial D.$ Clearly $x$ is not exterior to $D$. So suppose $x \in D_0\not=D_m$, one of the connected components of $D$. Then $x \in \overline{D_m} \cap D_0$, a contradiction. (For $D_0, D_m \subset \mathbb{C},$ open and disjoint, this cannot be.)
You are correct. The subtletly that the Wikipedia sketch overlooks, which you correctly include, is that you need to apply the maximum modulus principle to the connected component of the point where the maximum is attained, assuming the maximum is attained on the interior.