= Uniform-mesh Darboux criterion
{title2=$U(f,D_n)-L(f,D_n) o0$}
A bounded function on $[0,1]$ is <Riemann integrable> if and only if its <upper Darboux sum> minus its <lower Darboux sum> on the equal-length partition $D_n$ tends to zero. Sufficiency follows directly from the <Riemann integrability criterion>. Necessity follows by comparing with a fixed partition whose gap is small and using the <finite bad-cell estimate for Darboux sums>. The equal-length partitions need not be nested.
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