Separating sets by Darboux Baire 1 and approximately continuous functions
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