## Real Analysis Exchange

### 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.

#### Article information

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

Dates
First available in Project Euclid: 14 June 2012

https://projecteuclid.org/euclid.rae/1339694078

Mathematical Reviews number (MathSciNet)
MR1407262

Zentralblatt MATH identifier
0879.26005

#### Citation

Ciesielski, Krzysztof; Recław, Ireneusz. Cardinal invariants concerning extendable and peripherally continuous functions. Real Anal. Exchange 21 (1995), no. 2, 459--472. https://projecteuclid.org/euclid.rae/1339694078

#### References

• J. B. Brown, Totally discontinuous connectivity functions, Coll. Math., 23 (1971), 53–60.
• J. B. Brown, P. Humke, and M. Laczkovich, Measurable Darboux functions, Proc. Amer. Math. Soc., 102 (1988), 603–609.
• K. Ciesielski and A. W. Miller, Cardinal invariants concerning functions, whose sum is almost continuous, Real Analysis Exch., 20 (1994-95), 657–672.
• W. W. Comfort and S. Negrepontis, The Theory of Ultrafilters, Springer Verlag, New York, 1974.
• H. Fast, Une remarque sur la propriété de Weierstrass, Colloquium Mathematicum, 7 (1959), 75–77.
• R. G. Gibson, A property of Borel measurable functions and extendable functions, Real Analysis Exch., 13 (1987-88), 11–15.
• R. G. Gibson and F. Roush, The restrictions of a connectivity function are nice but not that nice, Real Anal. Exchange, 12 (1986-87), 372–376.
• W. J. Gorman III, The homeomorphic transformations of $c$-sets into $d$-sets, Proc. Amer. Math. Soc., 17 (1966), 826–830.
• M. Hagan, Equivalence of connectivity maps and peripherally continuous transformations, Proc. Amer. Math. Soc., 17 (1966), 175–177.
• T. Natkaniec, Almost Continuity,Real Analysis Exch., 17, (1991-92), 462–520.
• T. Natkaniec, Extendability and almost continuity, Real Analysis Exch., 21 (1995-96), 349–355.
• T. Natkaniec and I. Recław, Cardinal invariants concerning functions whose product is almost continuous, Real Analysis Exch., 20 (1994-95), 281–285.
• H. Rosen, Limits and sums of extendable connectivity functions, Real Analysis Exch., 20 (1994-95), 183–191.
• H. Rosen, Every Real Function is the Sum of Two Extendable Connectivity Functions, Real Analysis Exch., 21 (1995–96), 299–303.
• S. Ruziewicz, Sur une propriété des functions arbitraires d'une variable réelle, Mathematica, 9 (1935), 83–85.