I encounter the problem that I would like to extend the implicit function theorem (for real numbers) to a global version.

The classical implicit function theorem is given by the following: Assume $F: \mathbb{R}^{n+m} \to \mathbb{R}^m$ is a continuously differentiable function and assume there is some $(x_0,y_0) \in \mathbb{R}^{n+m}$ such that $F(x_0,y_0) = 0$ and such that the Jacobian matrix (with respect to $y$) at $(x_0,y_0)$ is invertible. Then, there exists an open set $U \subseteq \mathbb{R}^n$ around $x_0$ and a unique continuously differentiable function $g: U \to \mathbb{R}^m$ such that $g(x_0) = y_0$ and $F(x,g(x)) = 0$ for all $x \in U$.

However, I would like to obtain a global version of the theorem which guarantees that $F(x,g(x)) = 0$ for all $x \in \mathbb{R}^n$ and that $g$ is unique on the whole space $\mathbb{R}^n$. Does anyone know easy conditions for this and/or a citeable reference such as a book or a published paper?

Thanks in advance!


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