Abstract
A function f : X → Y between topological spaces is called σ-continuous (resp. ̄σ-continuous) if there exists a (closed) cover {Xn}n∈ω of X such that for every n ∈ ω the restriction f ↾ Xn 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)}.