Symmetrically Continuous Functions on Various Subsets of the Real Line

Abstract

We define symmetric continuity for functions defined on arbitrary subsets of $\mathbb{R}$. The main result is that when a symmetrically continuous function is defined on a measurable set (a set with the Baire property), then it is continuous almost everywhere (on a residual set, respectively). This generalizes the known result for functions defined on the whole real line.