Past exam of the mathematics course of the University of Cambridge / 2025 / ia / Paper 1 / 4E (Analysis I)

The Darboux–Riemann criterion says that a bounded function on $[a,b]$ is Riemann integrable exactly when, for every $\epsilon >0$, some partition $P$ satisfies $U(f,P)-L(f,P)<\epsilon$. Indeed, every lower sum is at most every upper sum. Taking the supremum of lower sums and infimum of upper sums, the criterion makes their difference smaller than every positive $\epsilon$, so they are equal; this common value is the Riemann integral.

Given $\epsilon >0$, choose a partition of $[a,d]$ whose upper-minus-lower sum is below $\epsilon$, and refine it by inserting $b,c$. The contribution from subintervals lying in $[b,c]$ is nonnegative and no larger than the total difference. Restricting the refined partition to $[b,c]$ therefore gives upper-minus-lower sum below $\epsilon$, so the criterion proves integrability there.