  We define a family of three related reducibilities, $\leq_T$, $\leq_{tt}$ and
$\leq_m$, for arbitrary functions $f,g:X\rightarrow\mathbb R$, 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 $j_\alpha:2^\omega\rightarrow \mathbb R$ 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.
