Functions that satisfy $f(x,z) = f(x,y) f(y,z)$ I am specifically looking for solutions that are NOT of the form: $f(x,y) = g(x)/g(y)$ since that is an obvious solution, as is $f(x,y) = 0$. I have a suspicion that there may be answers to this question that do not fall into these categories.
The functions don't necessarily have to be continuous or differentiable.
Edit: $f$ is not necessarily defined on $\mathbb{R}^2$, it can be defined on any space you like. Basically the equation deals with functions of two variables/arguments.
Edit #2: Domains are irrelevant to this question.

First, either the function is zero everywhere or nowhere.  If $f(a,b)=0$, then $f(a,b')=0f(b,b')=0$, and then $f(a',b')=f(a',a)f(a,b')=0$.  Since this is for any $(a',b')$, the function is everywhere zero.
Second, suppose the function is zero nowhere.  Now we have $\frac{f(x,z)}{f(x,y)}=f(y,z)$, is independent of $x$.  Define $g(t)=f(1,t)$.  We have $\frac{g(z)}{g(y)}=f(y,z)$.  Hence the other "obvious" solution you mention is the only other possibility.
