ON FUNCTIONS DETERMINED BY DENSE SETS

Abstract

We show that a few basic classes of lower semicontinuous functions on ${\mathrm{ℝ}}^n$ are densely recoverable. Specifically, we show that the sum of a convex and a continuous function, the difference of two convex and lower semicontinuous functions, a K-increasing function (where K is a cone of nonempty interior), and differences of K-increasing functions are all functions uniquely determined by their values on a dense set in ${\mathrm{ℝ}}^n$. Thus, sets of such functions of each type are densely recoverable sets. In general, the sum and difference of two densely recoverable sets of functions is shown to not be densely recoverable.