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I was reading the this paper on Global Implicit function theorem. I am unable to reason one of the assumptions in the following statement:

Let $f(\mathbf{x}, \mathbf{y}): U \times V \to W$ be a continuously differentiable function. Here $U$ denotes an open subset of $\mathbb{R}^n$, $V$ an open convex set of $\mathbb{R}^m$ such that $m \geq n$ while $W$ is assumed to be the whole of $\mathbb{R}^n$. Furthermore, the rank of the matrix $\partial f/ \partial \mathbf{y}$ is $n$ for all points $(\mathbf{x}, \mathbf{y})$ such that $f(\mathbf{x}, \mathbf{y}) = \mathbf{0}$. If in addition to these, the following conditions are true

  • For some $\mathbf{y}_0 \in V$, there is exactly one $\mathbf{x}_0 \in U$ such that $f(\mathbf{x}_0, \mathbf{y}_0) = \mathbf{0}$
  • For each $S \in A$, there is a $T \in B$ such that $\mathbf{y} \in S$, $\mathbf{x} \in U$ and $f(\mathbf{x}, \mathbf{y}) = \mathbf{0}$ imply $\mathbf{x} \in T$
  • Det$(\partial f/ \partial \mathbf{x}) \neq 0$ for all points $(\mathbf{x}, \mathbf{y})$ such that $f(\mathbf{x}, \mathbf{y}) = \mathbf{0}$

then there is a unique $g: V \to U$ such that $f(g(\mathbf{y}), \mathbf{y}) = \mathbf{0}$ for all $\mathbf{y} \in V$ and $g$ is continuously differentiable on $V$. In the above statement, $A$ refers to any family of compact subsets of the metric space $V$ such that for each compact set $C \in V$, there is $S \in A$ such that $C \subseteq S$ and similarly $B$ refers to any family of compact subsets of the metric space $U$ such that for each compact set $D \in U$, there is $T \in B$ such that $D \subseteq T$.

I don't quite understand why should $\partial f/ \partial \mathbf{y}$ have a rank $n$. I can see why Det$(\partial f/ \partial \mathbf{x}) \neq 0$ should hold but not the other thing. Even in the proof, they do not use the condition that $\partial f/ \partial \mathbf{y}$ has a rank $n$. Furthermore, even in the proof of implicit function theorem (local version), there is no such assumption. So I am not sure why this additional condition is necessary.

Any insight into this will be really helpful! Thank you.

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