Let $f : (0,1)^2 \rightarrow \mathbb R_0$ be a continuous symmetric function such that, for all $u<s<t$ in $(0,1)$, there exists $g(t,s) \in \mathbb R_0$ such that $f(t,u)=g(t,s)f(s,u)$.
Does that mean that we can separate the variables of the function i.e write $f(t,s)=f_1(t)f_2(s)$ for some functions $f_1,f_2$ for all $s<t$ in $[0,1]$ ? It feels so obvious but I cannot find a proof. If it is not true (which would surprise me), can we add a small hypothesis so we can separate the variables ?