Let $a \in E$ and let $U$ be an open set of $E$ containing $a$. Let $f : U \rightarrow E$ be an application of class $\mathscr{C}^1$ on $U$ such that $df(a) = \operatorname{Id}_E$. We fix a real number $r > 0$ such that $\overline{B(a,r)} \subset U$ and $$\forall x_1, x_2 \in \overline{B(a,r)}, \quad \left\|f(x_1) - f(x_2)\right\| \geqslant \frac{1}{2} \left\|x_1 - x_2\right\|.$$ We denote by $W = \left\{y \in E \left\lvert\, \|y - f(a)\| < \frac{r}{4}\right.\right\}$ and $V = f^{-1}(W) \cap B(a,r)$. Show that $$f_{\mid V} : \begin{array}{lcc} V & \longrightarrow & W \\ x & \longmapsto & f(x) \end{array}$$ is a continuous bijection from $V$ to $W$ whose inverse is a continuous function on $W$.
Let $a \in E$ and let $U$ be an open set of $E$ containing $a$. Let $f : U \rightarrow E$ be an application of class $\mathscr{C}^1$ on $U$ such that $df(a) = \operatorname{Id}_E$. We fix a real number $r > 0$ such that $\overline{B(a,r)} \subset U$ and
$$\forall x_1, x_2 \in \overline{B(a,r)}, \quad \left\|f(x_1) - f(x_2)\right\| \geqslant \frac{1}{2} \left\|x_1 - x_2\right\|.$$
We denote by $W = \left\{y \in E \left\lvert\, \|y - f(a)\| < \frac{r}{4}\right.\right\}$ and $V = f^{-1}(W) \cap B(a,r)$.
Show that
$$f_{\mid V} : \begin{array}{lcc} V & \longrightarrow & W \\ x & \longmapsto & f(x) \end{array}$$
is a continuous bijection from $V$ to $W$ whose inverse is a continuous function on $W$.