Discreteness of $F$-jumping numbers at isolated non-Q-Gorenstein points

We show that the $F$-jumping numbers of a pair $(X,\mathfrak{a})$ in positive characteristic have no limit points whenever the symbolic Rees algebra of $-K_X$ is finitely generated outside an isolated collection of points. We also give a characteristic zero version of this result, as well as a generalization of the Hartshorne-Speiser-Lyubeznik-Gabber stabilization theorem describing the non-$F$-pure locus of a variety.