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I'm having trouble understanding this theorem:

2-13 Theorem. Let $f: \mathbb{R}^n \to \mathbb{R}^p$ be continuously differentiable in an open set containing $a$, where $p \leq n$. If $f(a) = 0$ and the $p \times n$ matrix $(D_jf^i(a))$ has rank $p$, then there is an open set $A \subset \mathbb{R}^n$ containing $a$ and a differentiable function $h: A \to \mathbb{R}^n$ with differentiable inverse such that $$f \circ h(x^1, \ldots, x^n) = (x^{n-p+1}, \ldots, x^n).$$

It seems like the function $f(x) = x^2 - \frac{1}{4}$, around $a = -\frac{1}{2}$, serves as a counterexample. Obviously this function is continuously differentiable everywhere, and $f(a) = 0$, and the derivative $f'(x) = 2x$ is nonzero at $a$. Since the domain of $h$ is $A$ which contains $a$, there must be $h(a)$ such that $f(h(a)) = a$ by the statement of the theorem, i.e., $f(h(-\frac{1}{2})) = -\frac{1}{2}$, but this is impossible as $f$ attains its minimum at $(0, -\frac{1}{4})$.

I looked at the proof and it seems that there is an error here:

Proof. We can consider $f$ as a function $f: \mathbb{R}^{n-p} \times \mathbb{R}^p \to \mathbb{R}^p$. If $\det M \neq 0$, then $M$ is the $p \times p$ matrix $(D_{n_p+j}f^i(a))$, $1 \leq i, j \leq p$, then we are precisely in the situation considered in the proof of Theorem 2-12, and as we showed in that proof, there is $h$ such that $f \circ h(x^1, \ldots, x^n) = (x^{n-p+1}, \ldots, x^n)$.

The error is that "the situation considered in the proof of Theorem 2-12" establishes the existence of $h(x)$ where $x$ is in a neighbourhood of $(a^1, \ldots, a^{n-p}, 0, \ldots, 0)$ (by considering a function $F$ that replaces the last $p$ coordinates of $x$ by $f(x)$, then applying the inverse function theorem) rather than in a neighbourhood of $a$ itself.

What is the correct statement of theorem 2-13? I would just skip it, but it appears to become important later on, in the manifolds chapter.

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  • $\begingroup$ See this question for the answer. $\endgroup$ – al0 May 6 '13 at 21:04
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For completeness, the corrected version of the theorem is stated here:

$\textbf{Corrected Theorem 2-13}.$ Let $f:\mathbf{R}^n\to\mathbf{R}^p$, $p\leq n$, be continuously differentiable in an open set containing $a$. If $f(a)=0$ and $\mathrm{rk}([D_jf^i(a)])=p$, then there is an open set $A\subset\mathbf{R}^n$ with $a\in A$ and an invertible function $h:A\to\mathbf{R}^n$, $h\in C^1$ and $h^{-1}\in C^1$, such that $h(a)=0$ and $$f\circ h^{-1}(x^1,\ldots, x^n)=(x^{n-p+1},\ldots, x^n).$$

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