The Integral as a Function of the Interval
Let’s say we take an integrator of bounded variation on an interval and a function that’s Riemann-Stieltjes integrable with respect to over that interval. Then we know that is also integrable with respect to over the subinterval . Let’s use this to define a function on by
We can immediately say some interesting things about this function. First of all, is, like , of bounded variation. Next, wherever is continuous, so is . Finally, if is increasing, then is differentiable wherever is differentiable and is continuous. At such points, we have . Notice that, as usual, the first two results will hold if we show that they hold for increasing integrators.
These results are similar to those we get from the Fundamental Theorem of Calculus, and we can use some of the same techniques to prove them. In particular, we call on the Integral Mean-Value Theorem for Riemann-Stieltjes integrals. If we take points and in , this tells us that
where is some point between and .
Now we let be the supremum of on . For any partition of we calculate the variation
thus giving an upper bound to the variation of .
Similarly, let be a point where is continuous. Then given an we can find a so that implies that . Thus , and we see that is continuous at as well.
Finally, we can divide by to find
Then as gets closer to , gets squeezed in towards as well. If is continuous at and is differentiable there, then the limit of this difference quotient exists, and has the value stated.