Topological reducibilities for discontinuous functions and their structures

In this article, we give a full description of the topological many-one degree structure of real-valued functions, recently introduced by Day—Downey—Westrick. We also 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 has continuous Weihrauch rank α , then f ′ has continuous Weihrauch rank α + 1, where f ′( x ) is defined as the Turing jump of f ( x ).