  We show that in any $\mathbb{Q}$-Gorenstein flat family of klt singularities,
normalized volumes are lower semicontinuous with respect to the Zariski
topology. 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, Liu and Xu, we show that K-semistability is a
very generic or empty condition in any $\mathbb{Q}$-Gorenstein flat family of
log Fano pairs.
