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 $\beta$ and the convergence index $\gamma$. It is shown that these two indices are fully compatible in the following sense : a Baire-1 function $f$ satisfies $\beta (f)\le {\omega }^{{\xi }_1}\cdot {\omega }^{{\xi }_2}$ for some countable ordinals ${\xi }_1$ and ${\xi }_2$ if and only if there exists a sequence of Baire-1 functions $(f_n)$ converging to $f$ pointwise such that ${\sup }_n\beta (f_n)\le {\omega }^{{\xi }_1}$ and $\gamma ((f_n))\le {\omega }^{{\xi }_2}$. We also obtain an extension result for Baire-1 functions analogous to the Tietze Extension Theorem. Finally, it is shown that if $\beta (f)\le {\omega }^{{\xi }_1}$ and $\beta (g)\le {\omega }^{{\xi }_2},$ then $\beta (fg)\le {\omega }^{\xi },$ where $ξ=\max\{ξ_1+ξ_2, ξ_2+ξ_1}\}.$ These results do not assume the boundedness of the functions involved.