Riemann Integrability

Darboux upper and lower sums, and when they define an integral.

The idea

A function is Riemann integrable when the limit of sums that defines its integral actually exists. Darboux's criterion decides this without mentioning sample points at all, by measuring each slice from above and from below.

Definition (Upper and lower sums).

Fix a partition $P$ of $[a, b]$ into pieces of widths $\Delta x_{i}$, and on piece $i$ let $M_{i}$ be the supremum of $f$ and $m_{i}$ its infimum. The upper sum and lower sum of $f$ over $P$ are $U(f, P) = \sum_{i} M_{i} \, \Delta x_{i}, \qquad L(f, P) = \sum_{i} m_{i} \, \Delta x_{i}.$

Every Riemann sum for $P$, whatever its sample points, lies between $L(f, P)$ and $U(f, P)$, so every candidate value of the integral lies between them too. Refining the partition can only lower $U$ and raise $L$, so the two sums squeeze toward each other.

Theorem (Darboux criterion).

A bounded function $f$ is Riemann integrable on $[a, b]$ if and only if for every $\varepsilon > 0$ some partition $P$ achieves $U(f, P) - L(f, P) < \varepsilon$. The upper and lower sums then share a single value, $\sup_{P} L(f, P) = \inf_{P} U(f, P)$, and that number is $\int_{a}^{b} f$.

Ways to work on it

• Walkthrough. The Darboux criterion, an upper sum, and the Dirichlet non-example.
• Practice. Compute an upper or lower Darboux sum.
• Hardest. Integrate a simple function directly from the definition.

Not sure where to start? Take the ten-question placement test.