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Theorem: Let $F: X\subseteq \mathbb{R}^n \rightarrow \mathbb{R}$ be of class of $C^1$ and let $a$ be a point of the level set $S=\{x\in\mathbb{R}^n \mid F(x)=c\}$. If $F_{x_n}(a)\neq 0$ then there is a neighborhood $U$ of $(a_1,a_1\dots a_{n-1}) \in \mathbb{R}^{n-1}$, a neighborhood $V$ for of $a_n\in\mathbb{R}$ and a function $f:U\subseteq\mathbb{R}^{n-1} \rightarrow V$ of class $C^1$ such that if $(x_1,x_2,\ldots,x_{n-1})\in U$ and $x_n \in V$ satisfy $F(x_1,x_2,\ldots,x_n)=c$ ,then $x_n = f(x_1,x_2,\ldots,x_{n-1})$ is representable as a function of $(x_1,\ldots,x_{n-1})$ .

I don't quite get what the theorem is trying to show. Can someone explain it? Is it just as simple as we just consider different part of the function?

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    Two related threads: http://math.stackexchange.com/questions/26205/what-is-the-implicit-function-theorem and http://math.stackexchange.com/questions/34824/inverse-implicit-function-theorem-reasons2012-01-09

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