1. a. Verify that the relative extrema of functions in $S$ are strict. b. Let $f \in S$. Show that the restriction of $f$ to the closure of each component of $\mathbb{R} \backslash E(f)$ is strictly monotone. Deduce that if $x \in E(f) \backslash \{\operatorname{Max} E(f)\}$ is a relative maximum (resp. minimum), the smallest element $y$ of $E(f)$ satisfying $y > x$ is a relative minimum (resp. maximum). c. Let $f \in S_n$ with $n \geq 2$. We set $\mathscr{E}(f) = f(E(f))$. Let $\sigma_f$ be the element of $\Sigma_n$ defined by $$\sigma_f = \beta_{\mathscr{E}(f)} \circ f \circ \beta_{E(f)}^{-1}$$ Show that $\sigma_f \in \operatorname{MD}(n) \cup \operatorname{DM}(n)$.
1. a. Verify that the relative extrema of functions in $S$ are strict.\\
b. Let $f \in S$. Show that the restriction of $f$ to the closure of each component of $\mathbb{R} \backslash E(f)$ is strictly monotone. Deduce that if $x \in E(f) \backslash \{\operatorname{Max} E(f)\}$ is a relative maximum (resp. minimum), the smallest element $y$ of $E(f)$ satisfying $y > x$ is a relative minimum (resp. maximum).\\
c. Let $f \in S_n$ with $n \geq 2$. We set $\mathscr{E}(f) = f(E(f))$. Let $\sigma_f$ be the element of $\Sigma_n$ defined by
$$\sigma_f = \beta_{\mathscr{E}(f)} \circ f \circ \beta_{E(f)}^{-1}$$
Show that $\sigma_f \in \operatorname{MD}(n) \cup \operatorname{DM}(n)$.