KURSCHAK 2009 Problem 3

Find all functions $f:\mathbb{Z}\to \mathbb{Q}$ with the following properties: if $f(x)<c<f(y)$ for some rational $c$ , then $f$ takes on the value of $c$ , and $$f(x)+f(y)+f(z)=f(x)f(y)f(z)$$ whenever $x+y+z=0$ .