Maximum modulus theorem

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 ?

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.