1. f(a)≠0 for some a in R
2. f(xf(y))=yf(x) for all x,y in R

I realized after my post that there was 
a problem. I am working on a more complicated
function and maybe you can help me with two
questions.

1) Let x>0. Does the following exist?

   max { 2^i*3*j | i,j in Z and 2^i*3^j <= x }

2) If the answer to 1) is yes.

   Let x,y>0. If 

     2^m*3^n = max { 2^i*3*j | i,j in Z and 2^i*3^j <= x }

                       and

     2^p*3^q = max { 2^i*3*j | i,j in Z and 2^i*3^j <= y }

   does

     2^(m+p)*3^(n+q) = max { 2^i*3*j | i,j in Z and 2^i*3^j <= xy } ?