Given an arbitrary function of two variables, say $f(x_1,x_2)$, what are the conditions that $f$ has to satisfy such that $f(x_1,x_2)=g_1(x_1)g_2(x_2)$? And in general, $f(x_1,x_2,..,x_n)=g_1(x_1)g_2(x_2)...g_n(x_n)$ i.e. $f$ is completely multiplicatively separable.
Will $g_i$ be of any specific nature?
A possible test is $$\frac{f(x,a)}{f(x,b)}=\frac{g_2(a)}{g_2(b)}=\text{Cst}.$$
There are no restrictions on the $g_i$, as the product is always defined.