$x,y,z\in\mathbb R^3.$ $c:\mathbb R^2\to\mathbb R$. $a,b$ are constants.
Function $f(x,y,z)$ satisfies that $\forall z_1,z_2\in\mathbb R$, we have $$\{(x,y)||f(x,y,z_1)=f(x,y,z_2)\}=\{(x,y)||ax+by=c(z_1,z_2)\},$$
which intuitively means, for any $z_1,z_2$, the graph of $(x,y)$ is a line.
Given the information, what is a good decomposition for $f$?
By "decomposition", I mean to find a function form $F$, such that
$$f(x,y,z)=F(f_1(x),f_2(y),f_3(z),f_4(x,y),f_5(y,z),f_6(x,z)).$$
For example, one form that works is $f(x,y,z)=g(ax+by,z)$. But it is too strong and not necessary for our property.
Of course, you can assume the monotonicity of each $f_i$ or $F$, if needed. Each $f_i$ can remain unknown (and they might have to).
To be clearer, one could consider a two-dimensional case, $f(x,y)$. Define $g(x)=f(x,y_1)-f(x,y_2)+x$. Our property is equivalent of saying that the fix point of $g(x)$ is globally unique. We are generalizing the uniqueness of fix-point to multiple dimensions.
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