When can a regular surface be written as the graph of some function.

Question

I was reading this question and the second answer (written by Christian Blatter) begins by assuming that the surface $S$ can be written in the form $$(x,y) \mapsto (x,y,f(x,y))$$ for some function $C^1$ function $f$. This seems intuitively clear but I am unsure how to rigorously justify this.

Answer 1

We assume that the surface is given by an equation $F(x,y,z)=0$ where $F$ is of class $C^1$ and $\nabla F \neq (0,0,0)$ at the point $p_0$.

This means that at least one of the partial derivatives of $F$ is nonzero at $p_0$. By renaming the variables if necessary, we can assume that $\partial F/\partial z$ nonzero at $p_0$.

Then the hypotheses of the implicit function theorem are satisfied, and therefore the equation locally defines a $C^1$ function $z=f(x,y)$.

In other words, the surface can be parametrized by $x$ and $y$: $$(x,y,f(x,y)).$$