Let such that:
(1)
a) Prove that f is periodic
b) Prove that there are infinitely many functions such that (1) is true
In fact, veileen's method works nicely. Let . Then is the golden ratio, and satisfies the quadratic equation
The function f satisfies . Therefore
Now you can check (using the quadratic equation for ) that and . Thus and so
Edit. Looking again at veileen's comment, I notice that my solution would be much simpler if I had used the quadratic equation at each stage of the calculation, rather than saving it up until the end. In fact,
(because ),