For example: given a Riemann integrable function , how do we prove that ? It’s easy once we know that Riemann integrable functions have points of continuity, but this latter fact is not obvious at all.
Another version of this problem: prove that for every bounded strictly positive function the upper integral is strictly positive. (The upper integral is defined as the infimum of all upper Darboux sums over all finite partitions .) I can think only of using the Baire category theorem: one of the sets must be dense in some interval , hence every upper sum is bounded below by .
The versions of Fundamental Theorem of Calculus found in elementary first-course-in-analysis books do not go far beyond integrals of continuous functions. For example, “suppose is differentiable and is integrable…” can be replaced with “suppose ” without losing anything of importance. It would be reasonable then to limit the scope of integration to continuous functions.