2. We define a relation $\sim$ on $S$ as follows: for every pair $(f, g)$ of $S^2$, $f \sim g$ if and only if there exist two continuous bijections $\varphi : \mathbb{R} \rightarrow \mathbb{R}$ and $\psi : \mathbb{R} \rightarrow \mathbb{R}$, strictly increasing, which satisfy $f = \psi \circ g \circ \varphi$. a. Verify that $\sim$ is an equivalence relation on $S$ and show that each equivalence class of $\sim$ is contained in one of the sets $S_n, n \in \mathbb{N}$. b. Let $n \in \mathbb{N}^*$ and $\{u_1, \ldots, u_n\}, \{v_1, \ldots, v_n\}$ be subsets of $\mathbb{R}$ satisfying $u_1 < \cdots < u_n$ and $v_1 < \cdots < v_n$. Verify that there exists a continuous bijection $\chi : \mathbb{R} \rightarrow \mathbb{R}$ strictly increasing such that $\chi(u_k) = v_k$ for $1 \leq k \leq n$. c. Suppose that $f$ and $g$ are in $S_*$ and that $$\lim_{x \rightarrow \pm\infty} |f(x)| = +\infty, \quad \lim_{x \rightarrow \pm\infty} |g(x)| = +\infty$$ Prove that $f \sim g$ if and only if $\sigma_f = \sigma_g$. d. Does the preceding equivalence hold for two arbitrary functions $f$ and $g$ of $S_*$?
2. We define a relation $\sim$ on $S$ as follows: for every pair $(f, g)$ of $S^2$, $f \sim g$ if and only if there exist two continuous bijections $\varphi : \mathbb{R} \rightarrow \mathbb{R}$ and $\psi : \mathbb{R} \rightarrow \mathbb{R}$, strictly increasing, which satisfy $f = \psi \circ g \circ \varphi$.\\
a. Verify that $\sim$ is an equivalence relation on $S$ and show that each equivalence class of $\sim$ is contained in one of the sets $S_n, n \in \mathbb{N}$.\\
b. Let $n \in \mathbb{N}^*$ and $\{u_1, \ldots, u_n\}, \{v_1, \ldots, v_n\}$ be subsets of $\mathbb{R}$ satisfying $u_1 < \cdots < u_n$ and $v_1 < \cdots < v_n$. Verify that there exists a continuous bijection $\chi : \mathbb{R} \rightarrow \mathbb{R}$ strictly increasing such that $\chi(u_k) = v_k$ for $1 \leq k \leq n$.\\
c. Suppose that $f$ and $g$ are in $S_*$ and that
$$\lim_{x \rightarrow \pm\infty} |f(x)| = +\infty, \quad \lim_{x \rightarrow \pm\infty} |g(x)| = +\infty$$
Prove that $f \sim g$ if and only if $\sigma_f = \sigma_g$.\\
d. Does the preceding equivalence hold for two arbitrary functions $f$ and $g$ of $S_*$?