Stability of test ideals of divisors with small multiplicity

Let $(X, \Delta)$ be a log pair in characteristic $p>0$ and $P$ be a (not necessarily closed) point of $X$. We show that there exists a constant $\delta>0$ such that $\tau(X, \Delta)_P= \tau(X, \Delta + D)_P$ for each effective $\mathbb{Q}$-Cartier divisor $D$ with $\mathrm{mult}_P(D) <\delta$. As its application, we show that if $D$ is an $\mathbb{R}$-Cartier divisor on a strongly $F$-regular projective variety, then the non-nef locus of $D$ coincides with the restricted base locus of $D$. This is a generalization of a result of Musta\c{t}\v{a} to the singular case and can be viewed as a characteristic $p$ analogue of a result of Cacciola--Di Biagio.