Continuity of iteration and approximation of iterative roots

Motivated by computing iterative roots for general continuous functions, in this paper we prove the continuity of the iteration operators TnTn, defined by Tnf=fnTnf=fn. We apply the continuity and introduce the concept of continuity degree to answer positively the approximation question: If limm→∞Fm=Flimm→∞Fm=F, can we find an iterative root fmfm of FmFm of order nn for each m∈Nm∈N such that the sequence (fm)(fm) tends to the iterative root of FF of order nn associated with a given initial function? We not only give the construction of such an approximating sequence (fm)(fm) but also illustrate the approximation of iterative roots with an example. Some remarks are presented in order to compare our approximation with the Hyers–Ulam stability.