Upper and lower Darboux integrals

Definition

Let $a < b$ and let $f \colon [a, b] \to \mathbb{R}$ be bounded. The upper Darboux integral of $f$ is the infimum of the upper Darboux sums $U(f, P)$ over all partitions $P$ of $[a, b]$. The lower Darboux integral of $f$ is the supremum of the lower Darboux sums $L(f, P)$ over all partitions $P$.

Remarks

The lower Darboux integral is at most the upper one. The function is Darboux integrable if the two are equal. By a theorem, this is the same as Riemann integrability. For the Dirichlet function on $[0, 1]$, the upper integral is $1$ and the lower integral is $0$.

Explanations

L1 human

The upper integral is the best upper estimate of the area by step functions above the graph. The lower integral is the best lower estimate by step functions below the graph.