1. Let $n \in \mathbb{N}^*$. We denote Id the identity map of $\mathbb{R}^n$. We equip $\mathbb{R}^n$ with a norm denoted $\|\cdot\|$ and the space of linear maps from $\mathbb{R}^n$ to $\mathbb{R}^n$ with the associated norm, also denoted $\|\cdot\|$. For $x \in \mathbb{R}^n$ and $r \in \mathbb{R}^+$, we denote $B(x, r)$ (resp. $B(x, r]$) the open (resp. closed) ball with center $x$ and radius $r$. Let $\mathcal{O}$ be an open set of $\mathbb{R}^n$ containing 0 and let $f : \mathcal{O} \rightarrow \mathbb{R}^n$ be a map of class $C^1$ such that $f(0) = 0$ and whose differential $\varphi$ at 0 is invertible. a. We set $g = \mathrm{Id} - \varphi^{-1} \circ f$. Show that $g$ is of class $C^1$ on $\mathcal{O}$ and that there exists $\varepsilon > 0$ such that $B(0, \varepsilon) \subset \mathcal{O}$ and $\|Dg(x)\| \leq \frac{1}{2}$ for $x \in B(0, \varepsilon)$. Deduce that $f$ is injective in $B(0, \varepsilon)$. b. Let $0 < r < \varepsilon$ and let $z_0 \in B(0, r/2)$. We set $h(x) = g(x) + z_0$ for $x \in \mathcal{O}$. Show that $$h(B(0, r]) \subset B(0, r].$$ c. Show that there exists $a \in B(0, r]$ such that $f(a) = \varphi(z_0)$. d. Let $W = \varphi(B(0, r/2))$ and $V = f^{-1}(W) \cap B(0, \varepsilon)$. Show that $V$ and $W$ are open and that $f_{|V}$ is a homeomorphism from $V$ to $W$.
1. Let $n \in \mathbb{N}^*$. We denote Id the identity map of $\mathbb{R}^n$. We equip $\mathbb{R}^n$ with a norm denoted $\|\cdot\|$ and the space of linear maps from $\mathbb{R}^n$ to $\mathbb{R}^n$ with the associated norm, also denoted $\|\cdot\|$. For $x \in \mathbb{R}^n$ and $r \in \mathbb{R}^+$, we denote $B(x, r)$ (resp. $B(x, r]$) the open (resp. closed) ball with center $x$ and radius $r$. Let $\mathcal{O}$ be an open set of $\mathbb{R}^n$ containing 0 and let $f : \mathcal{O} \rightarrow \mathbb{R}^n$ be a map of class $C^1$ such that $f(0) = 0$ and whose differential $\varphi$ at 0 is invertible.\\
a. We set $g = \mathrm{Id} - \varphi^{-1} \circ f$. Show that $g$ is of class $C^1$ on $\mathcal{O}$ and that there exists $\varepsilon > 0$ such that $B(0, \varepsilon) \subset \mathcal{O}$ and $\|Dg(x)\| \leq \frac{1}{2}$ for $x \in B(0, \varepsilon)$. Deduce that $f$ is injective in $B(0, \varepsilon)$.\\
b. Let $0 < r < \varepsilon$ and let $z_0 \in B(0, r/2)$. We set $h(x) = g(x) + z_0$ for $x \in \mathcal{O}$. Show that
$$h(B(0, r]) \subset B(0, r].$$
c. Show that there exists $a \in B(0, r]$ such that $f(a) = \varphi(z_0)$.\\
d. Let $W = \varphi(B(0, r/2))$ and $V = f^{-1}(W) \cap B(0, \varepsilon)$. Show that $V$ and $W$ are open and that $f_{|V}$ is a homeomorphism from $V$ to $W$.