Oscillation of a function on an interval (${\mathrm{osc}}_{I}f={\sup }_{I}f-{\inf }_{I}f$)

For a bounded real function, its oscillation on an interval is its full range width there. An upper Darboux sum minus a lower Darboux sum is the sum of these widths multiplied by their interval lengths. This gives the Riemann integrability criterion in terms of oscillation.