I have 3 problems with Theorem 1.3.
Problem 1) I don't see how Theorem 7.1 applies here. I have provided the statement and the proof given by Serge Lang and all the other proofs used for Theorem 1.3.
Theorem 1.3 Statement:
Let $U$ be a connected open set, and let $f$ be an analytic function on $U$. If $z_0<U$ is a maximum point for $|f|$, that is $|f(z_0)|\geq|f(z)|$ for all $z∈U$, then $f$ is constant on $U$
By Theorem 7.1 we have $f$ locally constant at $z_0$. Then by Theorem 1.2 ii) (compare with the constant function and $f$) $f $ is constant on $U$
Theorem 7.1 Statement:
Let $f$ be analytic on an open set $U$. Let $z_0 ∈ U$ be a maximum for $|f|$, that is , $|f(z_0)|\geq|f(z)|$ for all $z∈U$. Then $f$ is locally constant at $z_0$
Theorem 1.2 ii) Statement:
Let $f,g,$ be analytic on $U$. Let $S$ be a set of points in $U$ which is not discrete. Assume that $f(z)=g(z)$ for all $z$ in $S$. Then $f=g=$ on $U$.
Problem 2) Is the difference between Theorem 1.3 (Maximum Modulus Principle) and Theorem 7.1(Local Maximum Modulus Principle) that in 1.3 we are talking about any open set but in Theorem 7.1 we are specifically talking about connected open set? Also if it is, I can't intietively see why in the local case the maximum can exist whereas in the connected case the maximum can not exist and the function must be constant
Problem 3) In the case of regular calculus it is obvious that Theorem 1.3 doesn't hold but shouldn't it since we have the set of real numbers beeing an open set that is connected?
Any help would be appreciated. If you wish for me to clarify something from the book please do so and I will respond as quickly as I can.