Non-monotonic iterative roots extended from characteristic intervals

It has been treated as a difficult problem to find iterative roots of non-monotonic functions. For some PM functions which do not increase the number of forts under iteration a method was given to obtain a non-monotonic iterative root by extending a monotone iterative root from the characteristic interval. In this paper we prove that every continuous iterative root is an extension from the characteristic interval and give various modes of extension for those iterative roots of PM functions.