Let $f:A\subset\mathbb {R}^n\to\mathbb{R}$ a $C^1$ function on $A$, and let $y\in\mathbb{R}$ such that $\nabla f(x)\neq 0$, for every $x\in f^{-1}(y)$. Show that in a neighborhood of each point $p\in f^{-1}(y)$, the set $f^{-1}(y)$ can be parametrized by a $C^1$ function $g:B\subset\mathbb{R}^{n-1}\to \mathbb {R}^n$ on $B$.

I did this: Let $x=(x_1,\ldots,x_n)\in f^{-1}(y)$, then $\nabla f(x)=\left(\dfrac{\partial f}{\partial x_1}(x), \ldots, \dfrac{\partial f}{\partial x_n}(x)\right)\neq 0$, then exists $i\in\{1,...,n\}$, such that, $\dfrac{\partial f}{\partial x_i}(x)\neq 0$. I feel like have to use the Implicit Function theorem, but I think that $g$ must be a real function, i.e., $g:B\to \mathbb{R}$.

Thank you


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