Integration by parts for Riemann–Stieltjes integrals

Let $a<b$ . If $f,g:[a,b]\to\mathbb R$ are of bounded variation and at least one is continuous, then both Riemann–Stieltjes integrals exist and

\int_a^b f\,dg \;+\; \int_a^b g\,df \;=\; f(b)g(b)-f(a)g(a).

Remarks

When $f$ and $g$ are continuously differentiable, substituting $df=f'(x)\,dx$ and $dg=g'(x)\,dx$ gives the usual integration-by-parts formula. The stated existence follows from the more general criterion that two bounded-variation functions with no common discontinuity are integrable with respect to one another.