Symmetric behavior in functions

Darji, Udayan B.

doi:10.1090/S0002-9939-1993-1172958-3

Symmetric behavior in functions
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by Udayan B. Darji

Proc. Amer. Math. Soc. 119 (1993), 915-923

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

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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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