Let $F$ be the cdf of a distribution. The distribution is not continuous and does not has a density function. Let $$g(x) = \int_{a(x)}^{b(x)}h(x, u)dF(u),$$ with $a$, $b$, $h$ continuous functions. As the pdf $f$ does not exist, what is the Leibniz Integral Rule in this condition?
Update: the simplified case can be useful $h(u,x) = x -y$.