On limitwise monotonicity and maximal block functions

We prove the existence of a limitwise monotonic function g : N→ N \ {0} such that, for any Π1 function f : N → N \ {0}, Ran f 6= Ran g. Relativising this result we deduce the existence of an η-like computable linear ordering A such that, for any Π2 function F : Q → N \ {0}, and η-like B of order type ∑ {F (q) | q ∈ Q }, B A . We prove directly that, for any computable A which is either (i) strongly η-like or (ii) η-like with no strongly η-like interval, there exists 0′-limitwise monotonic G : Q → N \ {0} such that A has order type ∑ {G(q) | q ∈ Q }. In so doing we provide an alternative proof to the fact that, for every η-like computable linear ordering A with no strongly η-like interval, there exists computable B ∼= A with Π1 block relation. We also use our results to prove the existence of an η-like computable linear ordering which is ∆3 categorical but not ∆ 0 2 categorical.