Regarding ln|f| being upper semicontinuous

Question

Let $f$ be a holomorphic function on the open unit disc $\mathbb{D}$ In $\mathbb{C}$. Can anyone tell why $\ln|f|$ is upper semicontinuous but not continuous? In particular $\ln|z|$, $z\in \mathbb{D}$.

Answer 1

$\{z: \ln |f(z)| < a\}=\{z: |f(z)| <e^{a}\}$ is open by continuity of $|f|$. So the function is upper semi-continuous.

$\ln |f|$ is not even finite valued in general, so it is not continuous. But it can be continuous in special cases. If $f(z)=e^{z}$ then $\ln |f(z)|=\Re z$ which is continuous.

More generally whenever $f$ is holomorphic and $f(z) \neq 0$ for all $z$ we get continuity. [This is because $f(z)=e^{g(z)}$ for some holomorphic function $g$ and $\ln |f(z)|=\Re g(z)$].