Let f:R→R satisfy

f(a)≠0 for some a in R

f(xf(y))=yf(x) for all x,y in R
Prove that f(x)=f(x) for all x in R.
Obviously f1(x)=x and f2(x)=1/x (with f2(0):=0) are solutions. They are pretty much the only solutions.
We have
f(f(x)*f(y))=y*f(f(x))=x*y
Applying f on both sides gives (due to f(f(x))=x, see my previous post)
(1) f(x)*f(y)=f(x*y)
From (1) it is easy to show that f(x^r)=f(x)^r for all rational numbers r. Next compute f(x) for all rational powers of 2: If x=2^r then f(x)=f(2)^r=f(2)^(log(x)/log(r))=x^(log(f(2))/log(2)).
So f(x)=x^c for some c. From f(f(x)) follows x=f(f(x))=f(x^c)=f(x)^c. Therefore c^2=1, i.e. c=1 or c=1.
So we have shown that f, when restricted to all rational powers of 2 (i.e. to x=2^r) is the identity or the reciprocal function.
Because the rational powers of 2 and their complement are dense in the positive real numbers, that means there can be no other solutions, except those that are discontinuous everywhere. I suspect those weird examples could be constructed easily.

Posted by JLo
on 20061229 08:44:43 