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A more consistent definition of tetration of tetration for rational exponents
#21
describes the function your assumption produces, i.e.:




and so on, while still allowing and retaining the property .

By the term orbit I'm refering to a term from dynamical systems where, given a point x, the sequence {x, f(x), f(f(x)), ...} is referred to as the orbit of f from x which is a way of referring to iteration without referring to the t in . By using it this way, though, I'm slightly misusing it, since its a sequence, and not a function. Here I'm using it as a function , sorry if it was confusing. One of the reasons why I like the term 'orbit' so much is that it pairs nicely with iterate which, given t, is a function . I've discussed these terms here as well.

Andrew Robbins
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#22
andydude Wrote: describes the function your assumption produces, i.e.:




and so on, while still allowing and retaining the property .

By the term orbit I'm refering to a term from dynamical systems where, given a point x, the sequence {x, f(x), f(f(x)), ...} is referred to as the orbit of f from x which is a way of referring to iteration without referring to the t in . By using it this way, though, I'm slightly misusing it, since its a sequence, and not a function. Here I'm using it as a function , sorry if it was confusing. One of the reasons why I like the term 'orbit' so much is that it pairs nicely with iterate which, given t, is a function . I've discussed these terms here as well.

Andrew Robbins

Ok, I see. Still a pathological function though. It cannot be made continuous at 1. If we define to conform with the limit of the tetraroots towards zero, then from the functional equation of tetration we must have:

.

However, the limit from the left of y=1 can be found by using the following:



Solving the latter with Maple, one gets , so there is a discontinuity at y=1. In short, no matter how this function is defined at y=0 and y=1, it cannot be made continuous simultaneously at y=0 and y=1.

I prefer it defined as:





and let it do as it pleases in between, even if it ends up discontinuous.
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