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 | References | Similar Articles | Additional Information

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 . We show that there is no such characterization of topological nature. We prove that given a zero-dimensional set , there exists a function whose set of points of symmetric continuity is topologically equivalent to . Thus, there is no "upper bound" on the topological complexities of . We also prove similar theorems about the set of points where a function may be symmetrically differentiable, symmetric, or smooth.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1993-1172958-3

Keywords:
Symmetric continuity,
symmetric derivative,
smooth,
perfect linearly independent sets,
the Baire property,
Borel functions,
coanalytic sets

Article copyright:
© Copyright 1993
American Mathematical Society