Open Access
1995/1996 Cardinal invariants concerning extendable and peripherally continuous functions
Krzysztof Ciesielski, Ireneusz Recław
Author Affiliations +
Real Anal. Exchange 21(2): 459-472 (1995/1996).


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.


Download Citation

Krzysztof Ciesielski. Ireneusz Recław. "Cardinal invariants concerning extendable and peripherally continuous functions." Real Anal. Exchange 21 (2) 459 - 472, 1995/1996.


Published: 1995/1996
First available in Project Euclid: 14 June 2012

zbMATH: 0879.26005
MathSciNet: MR1407262

Primary: 26A15 , 54A25
Secondary: 03E75

Keywords: cardinal invariants , extendable functions , functions with perfect road , peripherally continuous functions

Rights: Copyright © 1995 Michigan State University Press

Vol.21 • No. 2 • 1995/1996
Back to Top