This is probably a very stupid question, but here goes.
Consider a function $f(x,y)$ where $x,y\geq 0$. Suppose it is given that $$ \lim_{x \to 0} f(x, \lambda x) = g(\lambda) , \qquad \lambda > 0. $$ I would like to conclude from this that in a sufficiently small neighborhood of the origin, we have $$ f(x,y) = g \left( \frac{y}{x} \right) + O(x,y) . $$ where the $O(x,y)$ terms vanish when both $x$ and $y$ go to zero. I need these extra terms to be $O(x,y)$ and not just $o(1)$.
Is such a conclusion correct? If not, what additional assumptions would I need to make on $f$ to conclude this?