Real-analytic tetration uniqueness criterion?

[tommy1729]

Can you prove that if the property holds at \( x = A \), it also holds at \( x = A + B \) where \( B \) is between 0 and 0.5 ?
I believe that would be useful and could lead to induction which could prove the whole case.

Also is there a value \( C \) such that for \( x > C \) we have \( D^n tet(x) >= 0 \) for all positive integer \( n \) ?
Maybe Im confused ...

I once read about the same conditions as in the OP but for another function then \( tet(x) \). So there must be some theory I think.

I changed my mind, I think it might be true.

regards

tommy1729