




The "variation" of a function is the sum of its rises where it is increasing and its declines where it decreases.
In the previous comment, we proved:
Theorem 1: If f has bounded variation in an interval, its integral over that interval exists and is well defined.
A slightly stronger statement, also true, is:
Theorem 2: If f is continuous over a closed interval [a,b], it is integrable over it.