A classification of Baire class $1$ functions

We study in this paper various ordinal ranks of (bounded) Baire class $1$ functions and we show their essential equivalence. This leads to a natural classification of the class of bounded Baire class $1$ functions ${\mathcal{B}_1}$ in a transfinite hierarchy $\mathcal{B}_1^\xi (\xi < {\omega _1})$ of "small" Baire classes, for which (for example) an analysis similar to the Hausdorff-Kuratowski analysis of $\Delta _2^0$ sets via transfinite differences of closed sets can be carried out. The notions of pseudouniform convergence of a sequence of functions and optimal convergence of a sequence of continuous functions to a Baire class $1$ function $f$ are introduced and used in this study.