Formulas for translative functions

TL;DR

This paper explores the construction and characterization of translative functions on real vector spaces using Gerstewitz functionals, providing formulas and conditions for various continuity and topological properties.

Contribution

It introduces formulas for translative functions with specific properties and shows how any extended real-valued function relates to these translative functions.

Findings

Formulas for lower semicontinuous and continuous translative functions.

Characterization of continuity via epigraphs.

Representation of extended real-valued functions as restrictions of 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.