Functions of Baire class one

Abstract

Let $K$ be a compact metric space. A real-valued function on $K$ is said to be of Baire class one (Baire-1) if it is the pointwise limit of a sequence of continuous functions. In this paper, we study two well known ordinal indices of Baire-1 functions, the oscillation index $β$ and the convergence index $γ$. It is shown that these two indices are fully compatible in the following sense : a Baire-1 function $f$ satisfies $β(f) \leq ω^{ξ_1} \cdot ω^{ξ_2}$ for some countable ordinals $ξ_1$ and $ξ_2$ if and only if there exists a sequence of Baire-1 functions $(f_n)$ converging to $f$ pointwise such that $\sup_nβ(f_n) \leq ω^{ξ_1}$ and $γ((f_n)) \leq ω^{ξ_2}$. We also obtain an extension result for Baire-1 functions analogous to the Tietze Extension Theorem. Finally, it is shown that if $β(f) \leq ω^{ξ_1}$ and $β(g) \leq ω^{ξ_2},$ then $β(fg) \leq ω^ξ,$ where $ξ=\max\{ξ_1+ξ_2, ξ_2+ξ_1}\}.$ These results do not assume the boundedness of the functions involved.