Infinite tetration and superroot of infinitesimal

[Ivars]

Any 2 points on real axis are connected by a line (curve) which leaves real axis and goes to infinity and returns on next point, infinitesimally close to previous.

Ivars

What? This has nothing to do with what I am trying to say. Are you joking?

GFR

No... It is my intuitive understanding of continuity. You were talking about sinusoids in real plane connecting h even h odd , or perhaps I misunderstood?

I just took the sinusoids out of real xy plane. Based on the fact that tetration tends to take real numbers away from purely real , why can not h(x) < e^-e when extented to real tetration parameter (so that between n odd and n interger as n-> infinity we have n real such that as z-> infinity but is not integer, x^^z = imaginary or complex?