Suppose there is a CDF $F(t)$ that is allowed to have mass points. How can I calculate the right (or left) hand derivative of $\int_a^x f(x,t)dF(t)$. i.e. I want to get an expression for $$ \frac{\partial}{\partial x}_{\text{from the right}}\int_a^x f(x,t)dF(t) $$ The reason for such an approach is that I believe that the integral may not be differentiable (like if there are a countable number of mass points?). As such, I believe left/right derivatives are needed.
I understand that the right hand derivative is $$ \lim_{h\to 0^+} \frac{\int_a^{x+h}f(x+h,t)dF(t) - \int_a^xf(x,t)dF(t)}{h} $$ But I don't know where to go from here. Specifically, when I have limits of integrals (in a simple case like $\int f(x)dx$) then I evaluate the integral and take the limit. But Here I cannot evaluate the integral because $F(t)$ has mass points.
I would be satisfied with an answer assuming $f(x,t)$ is additively separable. For example, I would be happy to get an answer for $f(x,t) = (x-t)$, with $x\leq t$. I.e., you can answer for $$ \frac{\partial}{\partial x}_{\text{from the right}}\int_a^x (x-t)dF(t) $$
I believe that in this case the limit becomes $$ \lim_{h\to 0^+}\frac{\int_a^{x+h} (h)dF(t) + \int_x^{x+h}(x-t)dF(t)}{h} $$ but again, I don't know how to evaluate the limit to then take the limit (I do realize that $\lim_{h\to 0^+}F(s) = F(s)$)