Suppose I have a three-variable function $f(x_1, x_2, x_3)$, $f : \mathbb{R}^3 \to \mathbb{R}$.
If it is linear for $x_1$, $x_2$ and $x_3$ we can say it has the form $f(x_1, x_2, x_3) = c_1x_1 + c_2x_2 + c_3x_3$ where $c_1, c_2, c_3 \in \mathbb{R}$.
We can say it is linear for $x_1$ and $x_2$, for instance, iff it has the form $j(x_3)x_1 + z(x_3)x_2$ where $j,z : \mathbb{R} \to \mathbb{R}$.
I'm trying to understand how the concept of "affine" applies to multi-variable functions. Is there something as an affine-function for all variables? Given $f$ (in my previous case) were affine for all variables, would it's form be $j(x_3)x_1 + z(x_3)x_2 + h(x_3)$? Is there something as a function that's affine for only some of the variables? For example, what would be the form of $f$ if it were affine for $x_1$ and $x_2$?