3. We denote $C_b^0$ the space of continuous bounded functions from $\mathbb{R}$ to $\mathbb{R}$, equipped with the uniform norm: $\|f\| = \operatorname{Sup}_{x \in \mathbb{R}} |f(x)|$ for $f \in C_b^0$. a. Let $n \in \mathbb{N}^*, \{u_1, \ldots, u_n\} \subset \mathbb{R}$ and $\{v_1, \ldots, v_n\} \subset \mathbb{R}$ with $u_1 < \cdots < u_n$ and $v_1 < \cdots < v_n$. Show that there exists a continuous map $\zeta : [0,1] \times \mathbb{R} \rightarrow \mathbb{R}$ such that:
for $s \in [0,1]$, the function $x \mapsto \zeta(s, x)$ is a strictly increasing bijection from $\mathbb{R}$ to $\mathbb{R}$,
$\zeta(0, x) = x$ for $x \in \mathbb{R}$ and $\zeta(1, u_k) = v_k$, $1 \leq k \leq n$.
b. Prove that the equivalence classes of the restriction of $\sim$ to $S_* \cap C_b^0$ are arc-connected. c. Give an example of a continuous arc $\gamma : [0,1] \rightarrow S \cap C_b^0$ such that $\gamma(0) \in S_0$ and $\gamma(1) \in S_2$.
3. We denote $C_b^0$ the space of continuous bounded functions from $\mathbb{R}$ to $\mathbb{R}$, equipped with the uniform norm: $\|f\| = \operatorname{Sup}_{x \in \mathbb{R}} |f(x)|$ for $f \in C_b^0$.\\
a. Let $n \in \mathbb{N}^*, \{u_1, \ldots, u_n\} \subset \mathbb{R}$ and $\{v_1, \ldots, v_n\} \subset \mathbb{R}$ with $u_1 < \cdots < u_n$ and $v_1 < \cdots < v_n$. Show that there exists a continuous map $\zeta : [0,1] \times \mathbb{R} \rightarrow \mathbb{R}$ such that:
\begin{itemize}
\item for $s \in [0,1]$, the function $x \mapsto \zeta(s, x)$ is a strictly increasing bijection from $\mathbb{R}$ to $\mathbb{R}$,
\item $\zeta(0, x) = x$ for $x \in \mathbb{R}$ and $\zeta(1, u_k) = v_k$, $1 \leq k \leq n$.
\end{itemize}
b. Prove that the equivalence classes of the restriction of $\sim$ to $S_* \cap C_b^0$ are arc-connected.\\
c. Give an example of a continuous arc $\gamma : [0,1] \rightarrow S \cap C_b^0$ such that $\gamma(0) \in S_0$ and $\gamma(1) \in S_2$.