The Gale–Nikaido–Debreu lemma with discontinuous excess demand

We provide a generalization of the Gale–Nikaido–Debreu’s lemma for discontinuous excess demand in the light of recent work on the existence of equilibria in games with discontinuous payoffs. The standard upper hemicontinuity property of the excess demand is replaced by the weaker concept of “continuous inclusion property” introduced by He and Yannelis (J Math Anal Appl 450(2):1421–1433, 2017) and we allow for the cone Pof admissible prices to be general enough to cover cases for which commodities cannot be freely disposed of.