Function equation: $f(a,b)=f(a,c)+f(c,b)$ for all positive reals $a>c>b\geq 0$.
My solution:
$f(a,c)=f(a,b)-f(c,b)$,
Let $b=0$,
$f(a,c)=f(a,0)-f(c,0)$.
Define $g(a)=f(a,0)$. We get the answer.
It seems too simple; we don't even need the function be continuous or other conditions. Am I correct?