Upper Darboux integral ($\overline{{\int }_a^b}f={\inf }_{P}U(f,P)$)

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.