If $f(x) = x- \left \lfloor{x}\right \rfloor $ and $\alpha(x) = x^2$, show that f is Riemann-Stieltjes Integrable on [-1,2].
Every proof in my textbook seems to show that $\alpha$ is increasing on its domain and then shows f is RS-I, but that does not seem to be the case here. Since $\alpha$ is not increasing, how do I go about proving this?
$\alpha$ increasing is a very usual case, but being of bounded variation (difference between two monotone functions) is enough. Writing $\alpha$ as difference between two monotone functions is very easy in this case.
As $\alpha(x) = x^2$ is $C^1$, another property is applicable: $$\int_a^b fd\alpha = \int_a^b f\alpha'$$ where the RHS is simply a Riemann integral. In this case, the RHS exists because $f$ has a finite mumber of discontinuities.
See Riemann–Stieltjes integral