Are there any extensions of Riemann-Stieltjes integration that are able to handle the following integral?
$\int_0^1 \alpha \space d\alpha$
where
$ \alpha(x) = \left\{ \begin{array}{lr} 0 & x < \frac{1}{2} \\ \frac{1}{2} & x = \frac{1}{2} \\ 1 & x > \frac{1}{2} \end{array} \right. $
Everything I've found so far have strong requirements about the funcions' continuity, and are undefined when there are shared discontinuities, even for such a simple function.
My interest: Where integration by parts hold.
What consequence/uses would there be to it?