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Let $\rho :\mathbb{R}^N\longrightarrow\mathbb{R}$ be a continuously differentiable function such that $\rho (x) = 0 \Rightarrow d\rho (x) \not = 0$ for all $x\in\mathbb{R}^N$.

Suppose $\Omega \stackrel{\cdot}{=} \{ x\in\mathbb{R}^n;\ \rho (x)<0\}$ is non-empty and bounded. Use the Implicity Function Theorem to show that $\Omega$ is an open set with regular boundary.

I've tried this way: Since $\rho$ is continuous, $\Omega = \{x\in \mathbb{R}^N;\ \rho (x)<0\} = \rho^{-1} (]-\infty ,0[)$ is an open set. We also have that $x \in\partial{\Omega}\Leftrightarrow \rho (x)=0$ (is that ok?) and then, by hypothesis, $d\rho (x)\not = 0$ for all $x\in \partial{\Omega}$.

From this point, maybe use the implicit function theorem to define a parametrization for the set of points satisfying $\rho (x) = 0$, which I hope to solve the problem, but I got stuck.

Thanks.

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    You can't get the boundedness of $\Omega$ without additional assumptions: consider $\rho(x_1,x_2)=x_1^2-1$ in $\mathbb R^2$.2012-06-28

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