SYMMETRICAL BEHAVIOR IN FUNCTIONS

S. Marcus raised the following problem: Find necessary and sufficient conditions for a set to be the set of points of symmetric continuity o f some function f: R --> R. We show that there is no such characteriza tion of topological nature. We prove that given a zero-dimensional set M subset-or-equal-to R, there exists a function f: R --> R whose set of points of symmetric continuity is topologically equivalent to M . T hus, there is no ''upper bound'' on the topological complexities of M. We also prove similar theorems about the set of points where a functi on may be symmetrically differentiable, symmetric, or smooth.