Darboux sum

A lower Darboux sum uses the infimum of a function on each partition interval, while an upper sum uses the supremum.

For a bounded real function, its oscillation on an interval is its full range width there. An upper Darboux sum minus a lower Darboux sum is the sum of these widths multiplied by their interval lengths. This gives the Riemann integrability criterion in terms of oscillation.

For a bounded real function, inserting partition points increases its lower Darboux sum and decreases its upper Darboux sum. This follows interval by interval because infima increase on smaller sets while suprema decrease, and the new widths sum to the original width. A common partition refinement consequently proves $s(f,D_1)\le S(f,D_2)$ for arbitrary two partitions.

For a bounded function $f$ and a partition of an interval $P$, the upper Darboux sum multiplies the supremum of $f$ on each subinterval by its length and adds the results.

For a bounded function $f$ and a partition of an interval $P$, the lower Darboux sum multiplies the infimum of $f$ on each subinterval by its length and adds the results.

For a nonnegative Riemann-integrable function and an interval partition,Every factor is nonnegative, so multiplication of $1+t\le e^t$ is legitimate. The final step uses the lower-sum bound on the Riemann integral.

For a bounded function,The function is Riemann integrable exactly when these values agree.

The lower Darboux integral of a bounded real function on a closed interval is the supremum of its lower Darboux sums over all partitions of an interval. Refinement increases a lower Darboux sum. A common refinement of two arbitrary partitions of an interval proves that every lower Darboux sum is at most every upper Darboux sum, hence the lower Darboux integral never exceeds the upper Darboux integral.

The upper Darboux integral of a bounded real function on a closed interval is the infimum of its upper Darboux sums over all partitions of an interval. Refinement decreases an upper Darboux sum, and the function is Riemann integrable precisely when this infimum agrees with its lower Darboux integral.