Necessary and sufficient conditions in terms of lower cut sets are given for the insertion of a Baire-.5 function between two comparable real-valued functions on the topological spaces that F σ-kernel of sets are F σ-sets.

• 26A15 - Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
• 54C30 - Real-valued functions in general topology