I have a general scalar function which has the properties: \begin{align} f(s\,a,b,c)&=s\,f(a,b,c)\\ f(s\,a,s\,b,s\,c)&=f(a,b,c) \end{align}
where $s$ can be any real number, so the invariance is continous. How far can I restrict the function $f$ now? My approach is: $f$ is linear in $a$ and can be decomposed by a function $g$ that transforms as: \begin{align} f(a,b,c)&=a\cdot g(b,c)\\ g(s\,b,s \,c)&=\frac{1}{s} g(b,c) \end{align}
Can I further restrict $g$ from that tranformation law?
Especially can I assume any analytical form for $g$? If yes, how?
$g(b,c)$ will be a constant times $b^pc^q$, where $p+q=-1$
Note as mentioned in the comment, $g(b,c)=h(\frac bc)b^pc^q$. The function $h$ depends only on the ratio of the $b$ and $c$ numbers.