Symmetric behavior in functions

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