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

Substituting x=0 and y=0 gives f(0*f(0))=0*f(0) by condition 2.

We don't know what f(0) is yet, but it's clear that anything it could be times 0 must be 0. So it reduces to f(0)=0.