Limitwise monotonic sets of reals

1. f (x, s) f (x, s + 1) for all x and s; 2. sups f (x, s) < ∞ for every x ; 3. F(x) = sups f (x, s). A set A ⊆ N is limitwise monotonic if A equals to the range of some limitwise monotonic function. If we replace here the computable functions f by X -computable functions for some Turing oracle X then we get the notions of X -limitwise monotonic functions and sets, respectively. Note that a similar notion was introduced and studied in [5, 6] as an important description for computable abelian p-groups. In particular, a set is X -limitwise monotonic iff the abelian p-group ⊕ n∈A Zpn has an X -computable copy. For the purposes of our paper we could also note that the last is equivalent to the condition that the family of intervals n̂ = {i ∈ ω : i < n} where n ∈ A has an X -computable uniform enumeration. A survey on recent investigations in limitwise mononicity can be found in [1]. Together withMelnikov andKhoussainov [4], the second author studied the Turing degrees inwhich a particular set can be limitwise monotonic. In particular, the following result holds: Theorem 1.1 (Kalimullin-Melnikov-Khoussainov) Suppose a set S ⊆ N is limitwise monotonic in all noncomputable degrees. Then S is limitwise monotonic.