Solution
= Solution
The Darboux–Riemann criterion says that a <bounded function> on $[a,b]$ is Riemann integrable exactly when, for every $\varepsilon>0$, some partition $P$ satisfies $U(f,P)-L(f,P)<\varepsilon$. 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 $\varepsilon$, so they are equal; this common value is the Riemann <integral>.