Quasiconvexity and density topology

Abstract

We prove that if f : R^N --> R is quasiconvex and U is open in the density topology of R^N, then sup_U f = ess sup_U f, while inf_U f = ess inf_U f if and only if the equality holds when U = R^N. The first (second) property is typical of lsc (usc) functions and, even when U is an ordinary open subset, there seems to be no record that they both hold for all quasiconvex functions. This property ensures that the pointwise extrema of f on any nonempty density open subset can be arbitrarily closely approximated by values of f achieved on "large" subsets, which may be of relevance in a variety of issues. To support this claim, we use it to characterize the common points of continuity, or approximate continuity, of two quasiconvex functions that coincide away from a set of measure zero.