Title: On the semi-continuity problem of normalized volumes of singularities,
Abstract: We show that in any $\mathbb{Q}$-Gorenstein flat family of klt singularities,
normalized volumes can only jump down at countably many subvarieties. A quick
consequence is that smooth points have the largest normalized volume among all
klt singularities. Using an alternative characterization of K-semistability
developed by Li, Xu and the author, we show that K-semistability is a very
generic or empty condition in any $\mathbb{Q}$-Gorenstein flat family of log
Fano pairs.