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Consider the relation $f(x,y)=0$, with $f:\mathbb{R}^n\times\mathbb{R}^m\rightarrow \mathbb{R}^n$. The (standard) implicit value theorem gives you conditions for the existence of a function $g:B\rightarrow \mathbb{R}^m$ such that $f(x,g(x))=0$ for all $x$ in some open ball $B$ around a given point $a\in\mathbb{R}^n$. In addition, if $g$ exists then it inherits certain smoothness properties from $f$, i.e. if $f\in\mathcal{C}^k$ then $g\in\mathcal{C}^k$.

Suppose you already know there that there exists a $g$ such that $f(x,g(x))=0$ for all $x$ in a given open set A. What properties smoothness properties can $g$ be expected to have? More specifically if $f\in\mathcal{C}^k$ is $g\in\mathcal{C}^k$ and if $f$ is globally Lipschitz continuous is $g$ globally Lipschitz continuous?

Thanks in advance.

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    Sorry I meant $f'$ not $D_x f$.2012-08-22

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The implicit function theorem, assuming it's assumptions are satisfied, says there is a locally unique solution to the equation you wrote down. So you can expect exactly what the theorem claims, not more, not less, cause you get exactly what the theorem claims. In Siminores example the assumptions of the theorem are violated.

Oh, and I don't know of a Lipschitz version of the theorem and doubt there is one.

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    @copper.hat Right you are. But as far as I can tell, you wont have that 'ivertibilty in some sense' in general if $f$ is merely Lipschitz.2012-08-21