Separating sets by Darboux Baire 1 and approximately continuous functions

Abstract

We characterize sets A₀, A₁ for which there is a DB₁ function f with [f = 0] = A₀ and [f = 1] = A₁. This characterization is a conjunction of necessary conditions for Darboux and for Baire 1 functions. We also characterize sets A^- ,A⁺ for which there is a DB₁ function with [f < 0] = A^- and [f > 0] = A⁺. The same characterzations are provided for approximately continuous functions.

Keywords: Baire 1 function, Darboux function, separating sets, approximate continuity, Urysohn's lemma