σ-Continuous Functions and Related Cardinal Characteristics of the Continuum

A function f : X → Y between topological spaces is called σ-continuous (resp. ̄σ-continuous) if there exists a (closed) cover {X_n}_n∈_ω of X such that for every n ∈ ω the restriction f ↾ X_n is continuous. By 𝔠 _σ (resp. 𝔠_¯σ)we denote the largest cardinal κ ≤ 𝔠 such that every function f : X → ℝ defined on a subset X ⊂ ℝ of cardinality |X| <κ is σ-continuous (resp. ¯σ-continuous). It is clear that ω₁ ≤ 𝔠_¯σ ≤ 𝔠 _σ ≤ 𝔠.We prove that 𝔭 ≤ 𝔮₀ = 𝔠_¯σ =min{𝔠 _σ, 𝔟, 𝔮 }≤ 𝔠 _σ ≤ min{non(ℳ), non(𝒩)}.