Additivity, subadditivity and linearity: automatic continuity and quantifier weakening

TL;DR

This paper explores the relationships between additivity, subadditivity, and linearity, establishing automatic continuity results that simplify proofs in regular variation theory by reducing the need for complex arguments.

Contribution

It provides new automatic continuity theorems for additive and subadditive functions under minimal conditions, advancing the understanding of quantifier weakening in regular variation.

Findings

Additive functions are continuous under minimal regularity.

Subadditive functions are shown to be linear with minimal assumptions.

Reduces complex proofs in regular variation to simpler, more direct arguments.

Abstract

We study the interplay between additivity (as in the Cauchy functional equation), subadditivity and linearity. We obtain automatic continuity results in which additive or subadditive functions, under minimal regularity conditions, are continuous and so linear. We apply our results in the context of quantifier weakening in the theory of regular variation completing our programme of reducing the number of hard proofs there to zero.