A certain functional $T$ is defined as: $$T(F)=\int_{(0,1)}F^{-1}(s)M(ds)$$ where $M$ is a probability measure. The result that above functional is continuous at $F$, if the measure $M$ does not assign mass to the point where $F^{-1}$ is discontinuous is special case of Theorem 3.7. Robust Statistics by Huber and Ronchetti(2nd ed).

If $\frak{M}$ be a set of probability measures, is there a way to guarantee that $$T_{sup}(F)=\sup_{\{M\in\frak{M}\}}\{\int_{(0,1)}F^{-1}(s)M(ds)\}$$ is continuous if none of $M$ assign mass to point where $F^{-1}$ is discontinuous? Is it false?

We can at least say that it is lower semi continuous. Can we say more?