I'm quoting "Thomas' Calculus Early Transcendental 14th ed.":
In the plane, the points where a function of two independent variables has a constant value $f (x, y) = c$ make a curve in the function’s domain. In space, the points where a function of three independent variables has a constant value $f (x, y, z) = c$ make a surface in the function’s domain.
I have a very strong intuition that the above statement is true, but I don't know how to prove it though I did make some attempts and the author seems to take that for granted.