The Reflection of a Closed Interval

In the setting of Single-Variable Calculus on an Interval, let $a,b\in\mathbb{R}$ with $a<b$, so that $[a,b]$ is an interval containing the two distinct points $a$ and $b$, by Basic Facts about Intervals of the Real Line (v).

(i) There is exactly one map $\psi:[a,b]\to[a,b]$ with $\psi(u)=a+b-u$ for every $u\in[a,b]$.

In the remaining clauses, $\psi$ is this map.

(ii) For every $u\in[a,b]$, $\psi(\psi(u))=u$.

(iii) For all $u,v\in[a,b]$, $|\psi(u)-\psi(v)|=|u-v|$.

(iv) Read as a map from $[a,b]$ into $\mathbb{R}$, $\psi$ is continuous on $[a,b]$, as in The Real Line: Standing Notation and Background for Calculus on Intervals (vi).

(v) Read as a map from $[a,b]$ into $\mathbb{R}$, $\psi$ is differentiable, derivatives being taken relative to $[a,b]$, with derivative $\psi'(s)=-1$ at every $s\in[a,b]$, in the sense of Derivative of a Real Function on an Interval (ii).