Formulas for translative functions

Abstract

In this report, we consider extended real-valued functions on some real vector space. Gerstewitz functionals are used to construct all translative functions. We derive formulas for translative functions which are lower semicontinuous, continuous or have sublevel sets that are given by linear inequalities. Each extended real-valued function is shown to be the restriction of some translative function to a hyperspace. Continuity of an arbitrary extended real-valued function is characterized by its epigraph. Moreover, we study the directional closedness of sets as a base for the presented results.