Riemann integral is hard

For example: given a Riemann integrable function f\colon [a,b]\to (0,\infty), how do we prove that \int_{a}^b f(x)\,dx>0? 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 \overline{\int_{a}^b} f(x)\,dx is strictly positive. (The upper integral is defined as the infimum of all upper Darboux sums \displaystyle \sum_{j=1}^n \big\lbrace \sup_{[x_{j-1},x_j]} f \big\rbrace (x_j-x_{j-1}) over all finite partitions a=x_0<x_1<\dots<x_n=b.) I can think only of using the Baire category theorem: one of the sets \lbrace x\colon f(x)>1/m\rbrace must be dense in some interval I, hence every upper sum is bounded below by |I|/m.

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 F is differentiable and F' is integrable…” can be replaced with “suppose F\in C^1” without losing anything of importance. It would be reasonable then to limit the scope of integration to continuous functions.

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