Past exam of the mathematics course of the University of Cambridge / 2001 / ia / Paper 1 / 12D / Solution

For a bounded function on a nondegenerate compact interval $[a,b]$, a partition $P$ has Darboux sumsThe Riemann integrability criterion isEquivalently the supremum of lower sums and infimum of upper sums agree; their common value is the Riemann integral. A bounded nonintegrable example is the Dirichlet function $\boxed{f=1_{\mathbb{Q}}}$. Every positive-length interval contains rational and irrational points, so every lower sum is zero and every upper sum is $b-a>0$.