Intervals of Tetration

[UVIR]

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?

I checked it and it doesn't appear to work. Forcing each condition, \( C^2 \), \( C^3 \),...\( C^n \) gives separate and distinct solutions in x, so forcing all conditions simultaneously would give an overdetermined system and would thus be impossible. Back to the drawing board I guess.