Editing Maximum modulus principle (section)

===Related statement===

Suppose that <math>D</math> is a bounded nonempty connected open subset of <math>\mathbb{C}</math>.
Let <math>\overline{D}</math> be the closure of <math>D</math>.
Suppose that <math>f \colon \overline{D} \to \mathbb{C}</math> is a continuous function that is holomorphic on <math>D</math>.
Then <math>|f(z)|</math> attains a maximum at some point of the boundary of <math>D</math>.

This follows from the first version as follows. Since <math>\overline{D}</math> is [[compact space|compact]] and nonempty, the continuous function <math>|f(z)|</math> attains a maximum at some point <math>z_0</math> of <math>\overline{D}</math>. If <math>z_0</math> is not on the boundary, then the maximum modulus principle implies that <math>f</math> is constant, so <math>|f(z)|</math> also attains the same maximum at any point of the boundary.