Coanalytic sets and symmetric behavior in functions
type: Item
instance of: scholarly article
title: Coanalytic sets and symmetric behavior in functions
zbMATH Open document ID: 0812.26002
published in: Journal of Mathematical Analysis and Applications
publication date: 1995-05-07T00:00:00Z
review text: A function \(f: R\to R\) (\(R\) the real line) is said to be smooth (symmetric, symmetrically continuous) at a point \(p\) if \(\lim_{h\to 0} (f(p+h)+ f(p-h)- 2f(p))/ 2h=0\) \((\lim_{h\to 0} (f(p+h)+ f(p-h)- 2f(p))=0\), \(\lim_{h\to 0} (f(p+h)- f(p-h))=0)\). It is said that \(f^ s(p)\) is the symmetric derivative of \(f\) at \(p\) if \(\lim_{h\to 0} (f(p+h)- f(p-h))/ 2h= f^ s(p)\). If \(f\) is a function, then \(SM(f)\), \(S(f)\), \(SD(f)\), and \(SC(f)\) denote the set of points where \(f\) is smooth, symmetric, symmetrically differentiable, and symmetrically continuous, respectively. The author proves: Let \(M\) be a zero- dimensional coanalytic subset of \(R\). Then there exists a Baire 2 function \(f\) such that \(M\) is homeomorphic to \(SM(f)\), \(S(f)\), \(SD(| f|)\), and \(SC(| f|)\). This statement improves in some sense one of author's earlier results [Proc. Am. Math. Soc. 119, No. 3, 915-923 (1993; Zbl 0791.26002)].
Mathematics Subject Classification ID: 26A15
Mathematics Subject Classification ID: 26A24
zbMATH DE Number: 635941
zbMATH Keywords: symmetric functions
zbMATH Keywords: coanalytic sets
zbMATH Keywords: set of points
zbMATH Keywords: smooth
zbMATH Keywords: symmetric
zbMATH Keywords: symmetrically differentiable
zbMATH Keywords: symmetrically continuous
author: Udayan B. Darji
reviewed by: Pavel Kostyrko
MaRDI profile type: Publication
full work available at URL: jmaa.1994.1217
OpenAlex ID: W2081061035
DOI: 10.1006/JMAA.1994.1217
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description: scientific article; zbMATH DE number 635941