Let us say that a (total) function f:ℕ→ℕ from the natural numbers to the natural numbers is *behavior-preserving* when, for all p∈ℕ, the partial recursive functions φ_p and φ_{f(p)} defined by p and f(p) coincide (in the sense that they have the same domain of definition and the same values wherever they are defined). More informally: p and f(p) behave the same as programs.
Prove or disprove the following statement: for every decidable (=computable) set D⊆ℕ, either there exists a computable (total) behavior-preserving function whose image is contained in D, or there exists one whose image is contained in the complement ℕ∖D of D.