All solutions to a Schröder type functional equation: Schröder type functional equation: R. Mortini and R. Rupp.

We determine the solutions on various intervals in [ 0 , ∞ [ to the functional equation f (x m) = r f (x) for real r and positive m. Explicit formulas, involving periodic functions, are given for the set S of all solutions. The formulas for r < 0 are more complicated. An approach to S with the help of the axiom of choice is also given. A special attention is laid on solutions that are continuous on [ 0 , ∞ [ or on various open subintervals. We also describe solutions satisfying some asymptotic properties at the boundary of these intervals. [ABSTRACT FROM AUTHOR]