Let $V$ be an open subset of $\mathbb{R}^{n-1}$, $f\in C^1(V,\mathbb{R})$, and $S:=\{(x_1,\ldots,x_{n-1},f(x_1,\ldots,x_{n-1})) : (x_1,\ldots,x_{n-1}) \in V\}$ the graph of the function $f$. Now, is there a function $g\in C^1(\mathbb{R}^n)$ such that $S=g^{-1}(0)$. I think $0$ must be a regular value of $g$ but I cannot imagine what $g$ looks like.
Can someone help me please?