Is the Maximum Modulus Theorem Applicable if E2 is Open and Non-Constant?

[laurabon]

Homework Statement

Why the sets E1={z∈E:|f(z)|<M} and E2={z∈E:|f(z)|=M}} are both open?

Relevant Equations

E1={z∈E:|f(z)|<M}
E2={z∈E:|f(z)|=M}}

In the maximum modulus therem we have two sets $E1={z∈E:|f(z)|<M}E2={z∈E:|f(z)|=M}}$ I know that the set E1 is open because pre image of an open set. But should't be also E2 closed because pre image of only a point ?

 

[martinbn]

What is $f(z)$ and what is $E$? The first, my guess would be, is an analytic function. Also is this the definitions of $E_1$ and $E_2$? If $E$ is an open set and $E_2$ is defined by $\{z\in E\;:\;|f(z)|\le M\}$, then $E_2$ and $f$ is not constant, then $E_2$ is also open because the maximum is achieved on the boundary not in $E$. But these are just guesses. You should give all the information.