Let $f \colon \mathbb{C} \to \mathbb{R}_{+} \cup \left\{ 0 \right\}$ be a $C^{\infty}$ subharmonic function. Be given a compact domain $K \subset \mathbb{C}$, we let $D_{K}(f)$ be the set of critical points of $f$ lying in $K$, $V_{K}(f)$ be the set of critical values of $f \colon K \to \mathbb{R}_{+}$.
What can be said about $D_{K}(f)$ and $V_{K}(f)$? In particular, Do they have to be finite?
I am sorry if this question is obvious and something I haven’t given enough thought.
Thank you very much!