So where else have we seen derivatives as factors in integrands? Right! integration by parts! Here our formula says that
We can rewrite this using Riemann-Stieltjes integrals as
So if and are both continuously differentiable, this formula gives back our rule for integration by parts. But we can prove this without making those assumptions. In fact, we just need to assume that one of the two integrals exists, and the existence of the other one (and the formula) will follow.
Let’s assume that exists. That is, for every there is some tagged partition so that for every finer partition we have
Now let’s take any partition finer than and use it to set up the Rieman-Stieltjes sum
We can also use this partition to rewrite as
So subtracting the one from the other we find
But this is a Riemann-Stieltjes sum for the partition we get by throwing together all the and as partition points, and using as tags. This is a finer partition than , and so we see that
whenever is a partition finer than . This shows that the Riemann-Stieltjes integral of with respect to exists, and has the value we want.