Let $f:\mathbb{R}\times \mathbb{R}\mapsto \mathbb{R}$ , continous
$\forall$ $y$ in $\mathbb{R}$, $\exists$ $a,b$ such that $f(x,y) = ax + b$
$\forall$ $x$ in $\mathbb{R}$, $\exists$ $a',b'$ such that $f(x,y) = a'y + b'$
1/ does $a$,$b$,$a'$,$b'$ are the same for all $x$,$y$ ? Say :$\exists$ $a,b,a',b'$ such that $\forall$ $y$ in $\mathbb{R}$, $f(x,y) = ax + b$ and $\forall$ $x$ in $\mathbb{R}$, $f(x,y) = a'y + b'$
2/ is there a specific form for $f$ ?
NB : I am trying to resolve a real life unknow relation between two variables. I am able to compute any $f(x,y)$ but this is costly, I want to minimize the number of call on $f$.
Currently, I just called a few times $f$ for a given $y$ and found the above relation. Since I chose $y$ randomly, I am sure the relation is true for any $y$, but I am not sure of 1/ and I wonder if I can assert 1/ without additional computation on $f$
You could have $f(x,y) = a(y) x + b(y)$ for any continuous functions $a(y)$ and $b(y)$ on $\mathbb R$. No reason for them to be constant.
EDIT: With the new assumptions, you could have $f(x,y) = a x y + b x + c y + d$.