Topological reducibilities for discontinuous functions and their structures

In this article, we give a full description of a topological many-one degree structure of real-valued functions, recently introduced by Day-Downey-Westrick. We also point out that their characterization of the Bourgain rank of a Baire-one function of compact Polish domain can be extended to noncompact Polish domain. Finally, we clarify the relationship between the Martin conjecture and Day-Downey-Westrick's topological Turing-like reducibility, also known as parallelized continuous strong Weihrauch reducibility, for single-valued functions: Under the axiom of determinacy, we show that the continuous Weihrauch degrees of parallelizable single-valued functions are well-ordered; and moreover, if $f$ is has continuous Weihrauch rank $\alpha$, then $f'$ has continuous Weihrauch rank $\alpha+1$, where $f'(x)$ is defined as the Turing jump of $f(x)$.