# Basic help on Maximum Modulus Theorem

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!

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