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, 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.