A certain functional $T$ is defined as:
$$T(F)=\int\_{(0,1)}F^{-1}(s)M(ds)$$
where $M$ is a probability measure with support $[\alpha,1-\alpha]$,for $\alpha>0$.
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?


**Details Added:**


Domain of $T$ is CDFs. $F^{-1}(x):=\inf\{y\in\mathbb{R}:F(y)\ge x\}$


**Edit**


The integral should be $$T\_{sup}(F)=\sup\_{\{M\in\frak{M}\}}\{\int\_{[\alpha,1-\alpha]}F^{-1}(s)M(ds)\}$$ for $\alpha>0$



Continuity of a Functional
If all you assume is that none of the $M$ assigns mass to a point where $F^{-1}$ is discontinuous then the statement is certainly false. Assume you have a discontinuity at ${1/2}$, it suffices to consider the case where $\mathfrak{M}$ consists of a sequence of Dirac measures located at points that accumulate at $1/2$ from below... 


