Real Analysis Exchange

Cardinal invariants concerning extendable and peripherally continuous functions

Krzysztof Ciesielski and Ireneusz Recław

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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.

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Real Anal. Exchange, Volume 21, Number 2 (1995), 459-472.

First available in Project Euclid: 14 June 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 26A15: Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) {For properties determined by Fourier coefficients, see 42A16; for those determined by approximation properties, see 41A25, 41A27} 54A25: Cardinality properties (cardinal functions and inequalities, discrete subsets) [See also 03Exx] {For ultrafilters, see 54D80}
Secondary: 03E75: Applications of set theory

cardinal invariants extendable functions functions with perfect road peripherally continuous functions


Ciesielski, Krzysztof; Recław, Ireneusz. Cardinal invariants concerning extendable and peripherally continuous functions. Real Anal. Exchange 21 (1995), no. 2, 459--472.

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