Assume that ${\mathcal{X}}_{0}$ has klt singularities (and hence ${\mathcal{X}}$ is dlt by inversion of adjunction). Let $\psi$ be a continuous metric on $K_{{\mathcal{X}}/{\mathbb{D}}}$. Then $t\mapsto\int_{{\mathcal{X}}_{t}}e^{2\psi_{t}}$ is continuous at $t=0$.