= Solution
For a bounded function on a nondegenerate compact interval $[a,b]$, a partition $P$ has <Darboux sums>
$$
L(f,P)=\sum_i\inf_{[t_{i-1},t_i]}f\,(t_i-t_{i-1}),\qquad
U(f,P)=\sum_i\sup_{[t_{i-1},t_i]}f\,(t_i-t_{i-1}).
$$
The <Riemann integrability criterion> is
$$
\boxed{\forall\epsilon>0\ \exists P:\ U(f,P)-L(f,P)<\epsilon.}
$$
Equivalently 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 $b-a>0$.
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