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

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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 ?
 
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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.
 
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