Update
I'm trying to understand the author's proof of the Maximum Modulus Principle:
Let $f$ be analytic and non-constant in the domain $D$. Then $|f(z)|$ does not attain a maximum value at any point $z_0$ in $D$. In other words, there is no point $z_0$ in $D$ such that for all $z$ in $D$, $|f(z)|\le|f(z_0|$.
Now, I've followed the author's proof, understanding all the steps, then arriving at the result
$$|f(z_0)|=|f(z_0+re^{i\theta})|\quad\text{for all}\quad 0\le r\le R\quad\text{and}\quad 0\le \theta\le 2\pi$$
which means that $|f(z)|$ is a constant for all $z$ in $\overline{D}_R(z_0)$.
Now, by a previous theorem in the textbook, if the modulus of a function is constant on a closed disk, then the function is also constant in that closed disk, another theorem that I've proved.
Then he states that $f(z)=f(z_0)$ for all $z$ in the closed disk $\overline{D}_R(z_0)$.
This is the step I cannot understand. For example, why can't it be $f(z)=-f(z_0)$?
Thanks.