[FOM] Continuous Tetration

[joeshipman@aol.com]

What is the easiest way to explicitly define a 2-variable function from 
the non-negative reals to the non-negative reals with the following 
properties:

1) f(0,y)=y
2) f(1,y)=2^y
3) f(nx,y)=f(x,f((n-1)x,y)) for n=1,2,3,...
4) f is continuous
5) for all x, if y1>y2 then f(x,y1)>f(x,y2)

Although I believe an explicit and computable f satisfying 1-5 exists 
(and I think it ought to be possible to make f infinitely 
differentiable as well), I seem to remember reading that such a 
function cannot be analytic, and even that the one-variable restriction 
f(1/2,y) cannot be analytic (that is, there is no monotonic real 
analytic function f such that f(f(x))=2^x), but I'd appreciate seeing 
an explanation of this result.

-- Joe Shipman




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