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We show that in any $\mathbb{Q}$-Gorenstein flat family of klt singularities,normalized volumes can only jump down at countably many subvarieties. A quickconsequence is that smooth points have the largest normalized volume among allklt singularities. Using an alternative characterization of K-semistabilitydeveloped by Li, Xu and the author, we show that K-semistability is a verygeneric or empty condition in any $\mathbb{Q}$-Gorenstein flat family of logFano pairs.

On the semi-continuity problem of normalized volumes of singularities