Let $\alpha:[a,b]\rightarrow \mathbb{R}$ be a right continuous monotonically increasing function.
Let $f:[a,b]\rightarrow \mathbb{R}$ be a bounded function.
Let's say, $f$ is RS-integrable along $\alpha$ iff the Darboux upper sum and the Darboux upper sum conincide.
(Information: Using Lebesgue-Stieltjes measure (w.r.t $\alpha$), it can be shown that $f$ is RS-integrable iff the set of discontinuities of $f$ has measure zero)
Assume $f$ is RS-integrable along $\alpha$. Then, for a given $\epsilon>0$, does there exist $\delta >0$ such that for any partition $P=\{x_0,...,x_n\}$ of $[a,b]$, $mesh(P)<\delta \Rightarrow |\int_a^b f d\alpha - \sum_{i=1}^n f(\tau_i) (\alpha(x_i) - \alpha(x_{i-1}))|<\epsilon $ where $\tau_i$ is any value between $x_{i-1}$ and $x_i$?