Cardinal invariants concerning extendable and peripherally continuous functions

Abstract

Let \(\mathcal{F}\) be a family of real functions, \(\mathcal{F}\subseteq\mathbb{R}^\mathbb{R}\). In the paper we will examine the following question. For which families \(\mathcal{F}\subseteq\mathbb{R}^\mathbb{R}\) does there exist \(g\colon\mathbb{R}\to\mathbb{R}\) such that \(f+g\in\mathcal{F}\) for all \(f\in \mathcal{F}\)? More precisely, we will study a cardinal function \(\text{A}(\mathcal{F})\) defined as the smallest cardinality of a family \(F\subseteq\mathbb{R}^\mathbb{R}\) for which there is no such \(g\). We will prove that \(\text{A}(\text{Ext})=\text{A}(\text{PR})=\mathcal{c}^+\) and \(\text{A}(\text{PC})=2^{\mathcal{c}}\), where \(\text{Ext}\), \(\text{PR}\) and \(\text{PC}\) stand for the classes of extendable functions, functions with perfect road and peripherally continuous functions from \(\mathbb{R}\) into \(\mathbb{R}\), respectively. In particular, the equation \(\text{A}(\text{Ext})=\mathcal{c}^+\) immediately implies that every real function is a sum of two extendable functions. This solves a problem of Gibson \cite{Gib1}. We will also study the multiplicative analogue \(\text{M}(\mathcal{F})\) of the function \(\text{A}(\mathcal{F})\) and we prove that \(\text{M}(\text{Ext})=\text{M}(\text{PR})=2\) and \(\text{A}(\text{PC})=\mathcal{c}\). This article is a continuation of papers \cite{N2,CM,NR} in which functions \(\text{A}(\mathcal{F})\) and \(\text{M}(\mathcal{F})\) has been studied for the classes of almost continuous, connectivity and Darboux functions.