On limitwise monotonicity and maximal block functions

We prove the existence of a limitwise monotonic function g : N -> N \ {0} such that, for any Pi(0)(1) function f : N -> N \ {0}, Ran f not equal Ran g. Relativising this result we deduce the existence of an eta-like computable linear ordering A such that, for any Pi(0)(2) function F : Q -> N \ {0}, and eta-like B of order type Sigma{F(q) vertical bar q epsilon Q}, B not similar or equal to A. We prove directly that, for any computable A which is either ( i) strongly eta-like or (ii) eta-like with no strongly eta-like interval, there exists 0'-limitwise monotonic G : Q -> N \ {0} such that A has order type Sigma{G(q) vertical bar q epsilon Q}. In so doing we provide an alternative proof to the fact that, for every eta-like computable linear ordering A with no strongly eta-like interval, there exists computable B not similar or equal to A with Pi(0)(1) block relation. We also use our results to prove the existence of an eta-like computable linear ordering which is Delta(0)(3) categorical but not Delta(0)(2) categorical.