Three topological reducibilities for discontinuous functions

We define a family of three related reducibilities, $leq_T$, $leq_{tt}$ and $leq_m$, for arbitrary functions $f,g:Xrightarrowmathbb R$, where $X$ is a compact separable metric space. The $equiv_T$-equivalence classes mostly coincide with the proper Baire classes. We show that certain $alpha$-jump functions $j_alpha:2^omegarightarrow mathbb R$ are $leq_m$-minimal in their Baire class. Within the Baire 1 functions, we completely characterize the degree structure associated to $leq_{tt}$ and $leq_m$, finding an exact match to the $alpha$ hierarchy introduced by Bourgain and analyzed by Kechris and Louveau.