Derive $IF(x;T,F)$ when $$\displaystyle T(F)=\int_{F^{-1}(\alpha)}^{F^{-1}(1-\alpha)}x ~dF(x)$$ Here $IF$ stands for Influence function.
Trial: Here $$\begin{align}IF(x;T,F) &=\lim_{t\to 0}\frac{T((1-t)F+t\Delta_x)-T(F)}t \\ &=\lim_{t\to 0}\frac{g(t)-g(0)}t=\frac{d}{dt}g(t)|_{t=0} \end{align}$$ Then I try to simplify $T(F)$ as $$\int_{F^{-1}(\alpha)}^{F^{-1}(1-\alpha)}x ~dF(x) \\ =\int_{\alpha}^{1-\alpha}F^{-1}(y) ~dy$$ Then I am stuck. Please help.