Consider the following set up in real valued functions $x(y,z),y(z,x),z(x,y)$, all continuous and differentiable in a region $E$.
What I was thinking was that for $z(x,y)$, since $z$ is a function, and if one fix $x,y$ then $z$ would be fixed. Thus we obtain a surface of $z$, and this surface is all the place that $z$ is allowed to exist.
However, if consider $x(z,y)$, then there was another surface and constrines, and this is all the place that $x$ is free to be moving.
Since $x(y,z),y(z,x),z(x,y)$ are all functions, then for each function $x,y,z$, it's value takes on the interesction of two highly constrained surface, thus a line.
However, my professor gave me a funciton $x=yz$, where $x,y,z$ are all positive, and all $x(y,z),y(z,x),z(x,y)$ are surface, not lines.
What was worng with my original analysis?
$x=yz$ is not a function, it is an equation. It represents a smooth curved surface in $3-$ space. If you restrict the variables to be positive it is a single surface in the first octant of $\Bbb R^3$.
Given this surface, we can define a function $x(y,z)$, which will gives us $x$ if somebody gives us $y$ and $z$. Similarly we can define the other two functions because given any two coordinates we can derive the third. We can use it to define how fast $x$ changes with changes in $y$ by taking a partial derivative.