---
tag: 09FQT057T7
kind: definition
name: Upper and lower Darboux integrals
msc: [26A42]
defines: 0KABP2HWMQ
uses: [0QRQVH0CRW]
sources:
  - url: https://en.wikipedia.org/wiki/Darboux_integral
    locator: 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|0QRQVH0CRW]] $U(f, P)$ 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|021XJAQVDC]], this is the same as [[Riemann integrability|0N1JQXZXHC]].
For the [[Dirichlet function|03Z4JC4YRJ]] 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.