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 T n , defined by T n f = f n . We apply the continuity and introduce the concept of continuity degree to answer positively the approximation question: If lim m → ∞ F m = F , can we find an iterative root f m of F m of order n for each m ∈ N such that the sequence ( f m ) tends to the iterative root of F of order n associated with a given initial function? We not only give the construction of such an approximating sequence ( f m ) 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.