A surface in multivariable calculus typically represents one function of two independent variables, notated $z(x,\ y)$ or $z = f(x,\ y)$.
Suppose instead that $y$ is a function of $x$, and $z$ is still a function of $x$ and $y$, which we could notate as $z(x,\ y(x))$ or $y = f(x), z = g(x,\ y)$. Can we still represent $z$ as a surface, even though $y$ is not an independent variable in this case? If so, does this impose any additional limitations on what the geometry of the surface looks like?
Would this answer change if we allow $g(x,\ y)$ or $f(x)$ or both to be multi-functions?
This is a curve on a surface.
Specifically, it is the curve on the surface $z=g(x,y)$ over the graph of the curve $y=f(x)$ in the domain of $g$.