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. 
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 $\beta(f)\leq\omega ^{\xi_{1}}\cdot\omega^{\xi_{2}}$
for some countable ordinals $\xi_{1}$ and $\xi_{2}$ if and only
if there exists a sequence $(f_{n})$ of Baire-$1$ functions
converging to $f$ pointwise such that 
$\sup_{n}\beta(f_{n})\leq\omega^{\xi_{1}}$ and
$\gamma((f_{n}))\leq\omega^{\xi_{2}}$. We also obtain an
extension result for Baire-$1$ functions analogous to the Tietze
Extension Theorem. Finally, it is shown that if $\beta(f)
\leq\omega^{\xi_{1}}$ and $\beta( g) \leq\omega^{\xi_{2}}$, then
$\beta( fg) \leq\omega^{\xi}$, where $\xi=\max\{ \xi
_{1}+\xi_{2},\,\xi_{2}+\xi_{1}\} $. These results do not assume the
boundedness of the functions involved.

1. Denny H. Leung
   Department of Mathematics
   National University of Singapore
   2 Science Drive 2, Singapore 117543
   
2. Wee-Kee Tang
   Mathematics and Mathematics Education
   National Institute of Education
   Nanyang Technological University
   1 Nanyang Walk, Singapore 637616