a finite sequence of values such that

Each interval is called a subinterval of the partition. Let be a bounded function, and let

be a partition of . Let

The upper Darboux sum of with respect to is

The lower Darboux sum of with respect to is

The lower and upper Darboux sums are often called the lower and upper sums.

Darboux integrals

The upper Darboux integral of f is
:

The lower Darboux integral