Evaluate the limit $ \lim_{h\to 0}\frac{1}{h}\int_{u(t)}^{u(t+h)}f(x)dF(x) $

Question

I want to evaluate this limit of the following well defined Stieltjes integral: $$ \lim_{h\to 0}\frac{1}{h}\int_{u(t)}^{u(t+h)}f(x)dF(x) $$ Where:

• $u:\mathbb{R}\to \mathbb{R}$ is a strictly increasing function.
• $F$ and $f$ are real-valued bounded functions on $[u(t),u(t+h)]$.

I couldn't find an equivalence version of this result when the integral is with respect to Lebesgue measure which will be: $f(u(t))u'(t)$. But in this case I don't know what will I get ?! Any help will be appreciated! Thank you for your time :)