Cut norm discontinuity of triangular truncation of graphons

The space of Lp graphons, symmetric measurable functions w:[0,1]2→R with finite p-norm, features heavily in the study sparse graph limit theory. We show that triangular cut operator Mχ—the sets all values a graphon below main diagonal unit square to 0—acting on this is not continuous respect norm. This achieved by showing as n→∞, norm truncation Tn matrices equipped grows infinity well. Due density growth generalizes unboundedness Mχ. also