Let $g:[a,b] \to \mathbb{R}$ be a monotone function. Could you help me prove that $\mathcal{C}([a,b])\subseteq\mathcal{R}([a,b],g)$?
(Here $\mathcal{R}([a,b],g)$ is the set of all functions that are Riemann-Stieltjes integrable with respect to $g$.)
Definition of the Riemann-Stieltjes integral. Suppose $f,g$ are bounded on $[a,b]$. If there is an $A \in \mathbb{R}$ such that for every $\varepsilon >0$, there exists a partition $\mathcal{P}$ of $[a,b]$ such that for every refinement $\mathcal{Q}$ of $\mathcal{P}$ we have $|I(f,\mathcal{Q},X,g)-A|<\varepsilon$ (where if $\mathcal{P}=\{a=x_0<\ldots<x_n=b\}$ and $X$ is an evaluation sequence $X=\{x_1^\prime,\ldots,x_n^\prime\}$ and $I(f,\mathcal{Q},X,g)=\sum_{j=1}^n f(x_j^\prime)(g(x_j)-g(x_{j-1}))$), then $f$ is R-S integrable with respect to $g$, and the integral is $A$.
Best Answer
Assume that $g$ is increasing. I suppose that you know that $f\in\mathcal{R}([a,b],g)$ iff $f$ satisfies the Riemann's condition. The Riemann's condition says:
$f$ satisfies the Riemann's condition respect to $g$ in $[a,b]$ if for every $\epsilon\gt 0$, there exist a partition $P_\epsilon$ of $[a,b]$ such that if $P$ is a refinement of $P_\epsilon$ then $$0\leq U(P,f,g)-L(P,f,g)\lt \epsilon,$$ where $U(P,f,g)$ and $L(P,f,g)$ are the upper and lower Riemann-Stieltjes sums respectively.
With this and the hint in the comments the result holds.
Perhaps the chapter 7 of Mathematical Analysis of Tom M. Apostol can be useful to you.