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I was reading on Wikipedia that

"The maximum modulus principle can be viewed as a special case of the open mapping theorem, which states that a nonconstant holomorphic function maps open sets to open sets. If $|f|$ attains a local maximum at $a$, then the image of a sufficiently small open neighborhood of $a$ cannot be open. Therefore, $f$ is constant".

Could someone expand upon that? I don't follow why the image of a open neighborhood of a would not be open?

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    Because it can't contain an open neighborhood of $|f(a)|$ by maximality. (Here you need to observe that the absolute value function is also open away from the origin, but this is not hard to see.)2011-08-10

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