Integration by parts for Riemann–Stieltjes integrals

Integration by parts (Riemann–Stieltjes): Let $a<b$. Assume $f,g:[a,b]\to\mathbb{R}$ are of bounded variation and that at least one of $f$ or $g$ is continuous. Then both Riemann–Stieltjes integrals $\int_a^b f\,dg$ and $\int_a^b g\,df$ exist, and

This generalizes the usual integration by parts (obtained by taking $g(x)=x$ and interpreting $df=f'(x)\,dx$), and existence is ensured by the Riemann–Stieltjes integrability theorem .