Upper and lower Darboux integrals

For a bounded function,

\overline{\int_{a}^{b}} f = in f_{P} U (f, P), \underline{\int_{a}^{b}} f = sup_{P} L (f, P) .

The function is Riemann integrable exactly when these values agree.

Table of contents
- Lower Darboux integral Upper and lower Darboux integrals
- Upper Darboux integral Upper and lower Darboux integrals

Lower Darboux integral ( $\underline{\int_{a}^{b}} f = sup_{P} L (f, P)$ )

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.

Upper Darboux integral ( $\overline{\int_{a}^{b}} f = in f_{P} U (f, P)$ )

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.

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