Intervals of Tetration

[bo198214]

I *think* (but I am not absolutely sure about it) that any method which defines \( {^y}x=f(x,y) \) for reasonable \( f(x,y) \), for \( y\in [0,1] \) could conceivably be extended to a \( C^{\infty} \)

I dont get this. If I define it on \( (0,1) \) and I demand that \( {^{x+1}b}=b^{^xb} \) then the function is already determined for all \( x>0 \) there is no place left for further manipulations (i.e. to make it differentiable).

Andrew's approach was to choose it as an analytic function on \( (0,1) \) and he determines the coefficients of the seriesexpansion at 0 by demanding that it is \( C^{\infty} \) (or even analytic). However this approach works computably only on the \( \text{slog} \), the inverse of \( {^xb} \).