A function of two variables question Consider a function of two variables $h(x,y)$.    
If it's linear for $y$. Can I express it as $a(x)y + b(x)$ ?   If positive, why ?

Depending on your source, "linear for $y$" may mean different things. It may mean that $h$ is a linear map in the variable $y$, so that $$h(x,y_1+y_2)=h(x,y_1)+h(x,y_2)$$ and $$h(x,\alpha\cdot y)=\alpha\cdot h(x,y)$$ whenever both sides make sense. In that case--and if $h$ is a function $\Bbb R^2\to\Bbb R$, say--we do, indeed, have $$h(x,y)=a(x)\cdot y+b(x),$$ with $a(x)=h(x,1)$ and $b(x)\equiv 0$.

It could also be meaning to denote an affine map, in which case--again assuming $h:\Bbb R^2\to\Bbb R$--we have $$h(x,y)=a(x)\cdot y+b(x),$$ with $a(x)=h(x,1)$ and $b(x)=h(x,0)$.
