Let $$ f(x) = \begin{cases} 0 & \text{if $x=0$ or $x = 1/n$ for some $n \in \mathbb{N}$,} \\ 1 & \text{otherwise.} \end{cases} $$ Is this function Riemann-Stieltjes integrable in $[0,1]$?
For the upper Riemann-Stieltjes integral, all $f(x)$ would be $1$ for any partition so it's $1$.
For the lower Riemann-Stieltjes integral, for some $n \in \mathbb{N}$ of which $f(1/n)=0$, is there a chance of this integral becoming different from upper integral?
Your function is discontinuous in only countable number of points, hence it's Riemann integrable. Intuitively you can make the partition such that it's very small around the discontinuity points.
Refer to the link below
Proof that a function with a countable set of discontinuities is Riemann integrable without the notion of measure