In Analysis II by Tao, he wrote:
Theorem 6.7.2 (Inverse function theorem).
Let $E$ be an open subset of $\mathbf{R}^n$, and let $f : E \to \mathbf{R}^n$ be a function which is continuously differentiable on $E$. Suppose $x_0 \in E$ is such that linear transformation $f'(x_0) : \mathbf{R}^n \to \mathbf{R}^n$ is invertible. Then there exists an open set $U$ in $E$ containing $x_0$, and an open set $V$ in $\mathbf{R}^n$ containing $f(x_0)$, such that $f$ is a bijection from $U$ to $V$. In particular, there is an inverse map $f^{-1} : V \to U$. Furthermore, this inverse map is differentiable at $f(x_0)$, and $$ (f^{-1})'(f(x_0)) = (f'(x_0))^{-1}. $$
From errta of this book on his blog, he wrote:
Exercise 6.7.4. Let the notation and hypotheses be as in Theorem 6.7.2. Show that, after shrinking the open sets U, V if necessary (while still having $x_0 \in U$, $f(x_0) \in V$ of course), the derivative map $f'(x)$ is invertible for all $x \in U$, and that the inverse map $f^{-1}$ is differentiable at every point of $V$ with $(f^{-1})'(f(x)) = (f'(x))^{-1}$ for all $x \in U$. Finally, show that $f^{-1}$ is continuously differentiable on $V$.
I try to solve this exercise by looking for the relation between $f'(x)$ and $f'(x_0)$ for $x$ near $x_0$, but so far I got nothing. I also try to google proofs, turn out the exercise I'm tring to proof is given as a part of the hypothesis.
To conclude, I am looking for the answers of the following questions:
- How to proof the exercise for functions $f : \mathbf{R}^n \to \mathbf{R}^n$? (Tao already explained the case $f : \mathbf{R} \to \mathbf{R}$.) More specifically, proof the exercise without using some prior knowledge in linear algebra just like the way Tao proof the inverse function theorem (like determinants, etc. I did not learn linear algebra before).
- How to interpret the result of the exercises (with graph) ?
- Which parts of the given hypotheses are necessary?