# 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.) 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.