Is true that for any continuous function $g$ at $(x,y)$, there exists a function $f(x+y)$ such that
$$g(x,y)=a f(x+y)$$
with $a$ being a real scalar?
In other words, is it always possible to find a function $f$ (having only one argument $=x+y$) and it represents another function $g$ (which has two arguments $x$ and $y$) up to a scalar multiple?
I am not sure if it is always possible, sometimes possible, or never possible. Also I am not sure if this is a kind of translation.
For example, if $g(x,y)=x^2+xy+y^2$, what should be $a$ and $f$?
Your help would be really appreciated. THANKS!