Intervals of Tetration

[bo198214]

In other words, the tetration function defined by:

\( f(1.855276959,y)=1.855276959^y \) in (0,1) and by \( ^{y+1}1.855276959=1.855276959^{({^y}1.855276959)} \) is \( C^2 \) at y=1, assuming the way it was constructed by joining at the naturals with frac{y} as with my first construction.

Similar calculations show that using \( f(1.148776058,y) \) the corresponding function is \( C^6 \) at y=1, etc. I think one can go as high as one wants, provided a solution for x exists. It looks as if a solution \( x>1 \) exists for any even order derivative.

What do you guys think?

I didnt verify it directly but you dont need only the second derivative or only the 6th or only the \( n \)th derivative to be continuous for \( x\mapsto f(b,x) \) being in \( C^n \) but also all previous derivations. This means you get an equation system of at least \( n \) equations, but in only one variable.

Did you prove that in your case also the first or the first till fifth derivative is continuous?