Integration by parts (Riemann–Stieltjes)

Integration by parts (Riemann–Stieltjes): Let $f,g:[a,b]\to\mathbb{R}$ be functions of bounded variation, and assume at least one of $f$ or $g$ is continuous on $[a,b]$. Then both Riemann–Stieltjes integrals $\int_a^b f\,dg$ and $\int_a^b g\,df$ exist and $\int_a^b f\,dg + \int_a^b g\,df = f(b)g(b)-f(a)g(a).$

This identity generalizes the usual integration by parts formula for Riemann integrals and is essential in applications of the Riemann–Stieltjes integral (e.g., summation by parts and Fourier analysis).