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I have some difficulties understanding the Maximum Modulus Theorem, which seems counter-intuitive to me for now. Here is an example I have in mind:

Let's take $f(z) = 1 - (x^2+y^2)$ for $z = x +iy \in D = \{z \in \mathbb{C} / |z| < 1\}$. If I'm not mistaken, $f$ is holomorphic in $D$ and $D$ is open. However, $|f(0)| = 1 > |f(z)| = 0$ for $z$ on the boundary of $D$. This is against the theorem.

I read from this answer that in a neighborhood $U$ of $0$, $|f|(U)$ should be open, while here it's not the case. But again, I can't get why.

What am I missing? Thank you!

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You are missing the fact that the function $f$ is not holomorphic (actually, it is differentiable at $0$ and only at $0$) and that therefore the maximum modulus theorem doesn't apply here.

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  • $\begingroup$ I managed to find something about the differentiability of $|z|^2$, now I understand it. Thank you very much! $\endgroup$ May 25, 2019 at 10:59

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