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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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    Let $f(x,y)=1-y^2$ and $g(x)=-1$ when $x <0$, $g(x)=+1$ when $x \geq 0$. Then $f(x,g(x))=0$ for every $x$.2012-08-21
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    Generally some sort of invertibility is required. This is what constrains $g$ sufficiently to give it desirable properties.2012-08-21
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    Take $f(x,y) = x-\arctan y$. Then $f$ is smooth and globally Lipschitz, but an inverse function is not globally Lipschitz.2012-08-21
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    @copper.hat thanks, you're right in the end I'm getting requirements on the invertibility of different Jacobians.2012-08-21
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    For example, assume $f\in\mathcal{C}^1$. Then $f(x,g(x))=0$, so $f'(x,g(x)) = 0$, $D_x f(x,g(x)) = \frac{\partial f}{\partial x}+ \frac{\partial f}{\partial g}\frac{\partial g}{\partial x}=0\Rightarrow \frac{\partial g}{\partial x} = \left(\frac{\partial f}{\partial g}\right)^{-1}\frac{\partial f}{\partial x}$. Hence $g\in\mathcal{C}^1$ if and only if $\frac{\partial f}{\partial g}$ is invertible.2012-08-21
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    Sorry I meant $f'$ not $D_x f$.2012-08-22

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