I'm currently doing a question proving that if $f$ is a monotone function and $\alpha$ is a continuous and increasing function, then $f$ is Riemann-Stieltjes integrable.
I know that when $f$ is monotone the $$\sum_{i=1}^{n}(M_i -m_i)(\alpha(x_i)-\alpha(x_{i-1}))= (\sup(f)-\inf(f))((\alpha(x_i)-\alpha(x_{i-1}))$$
My question is, are there any other cases, other than $f$ being monotone, where this is true?