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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Symmetric behavior in functions


Author: Udayan B. Darji
Journal: Proc. Amer. Math. Soc. 119 (1993), 915-923
MSC: Primary 26A15; Secondary 26A24, 54C08
MathSciNet review: 1172958
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Abstract: S. Marcus raised the following problem: Find necessary and sufficient conditions for a set to be the set of points of symmetric continuity of some function $ f:R \to R$. We show that there is no such characterization of topological nature. We prove that given a zero-dimensional set $ M \subseteq R$, there exists a function $ f:R \to R$ whose set of points of symmetric continuity is topologically equivalent to $ M$. Thus, there is no "upper bound" on the topological complexities of $ M$. We also prove similar theorems about the set of points where a function may be symmetrically differentiable, symmetric, or smooth.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1993-1172958-3
PII: S 0002-9939(1993)1172958-3
Keywords: Symmetric continuity, symmetric derivative, smooth, perfect linearly independent sets, the Baire property, Borel functions, coanalytic sets
Article copyright: © Copyright 1993 American Mathematical Society