R is the reals. For what real k can we find a continuous function f : R → R such that f(f(x)) = k x9 for all x.
Solution
Evidently k1/4 x3 works for any k ≥ 0.
For k ≠ 0, f must be (1, 1) [otherwise we would get x9 = y9 for x ≠ y, contradiction.] But f is continuous, so it is strictly monotonic. If it is strictly monotonic increasing, then so is f(f(x)). But if it is strictly monotonic decreasing, then f(f(x)) is strictly monotonic increasing. So, either way, f(f(x)) is strictly monotonic increasing. If k < 0, then k x9 is strictly monotonic decreasing, so we cannot have k < 0.
© John Scholes
jscholes@kalva.demon.co.uk
4 Dec 1999