What is an order reversing bijection on $\Bbb R$?

Question

What is an order reversing bijection on $\Bbb R$ and why must it be continuous? Thanks

Answer 1

An order reversing bijection on $\mathbb{R}$, the set of real numbers, is a function $f \colon \mathbb{R} \to \mathbb{R}$ which is a bijection, that is, for every $y \in \mathbb{R}$, there is a unique $x \in \mathbb{R}$ with $f(x) = y$, and is order reversing, that is, for every $x_1<x_2 $, $f(x_1) > f(x_2)$.

To see that this is a continuous map (in fact a homeomorphism, since the inverse of $f$ will have the same properties), show that $f^{-1}(y_1,y_2) = (f^{-1}(y_2),f^{-1}(y_1))$.

Answer 2

A bijection $f: \mathbb R \to \mathbb R$ is said to be order reversing if $\forall x, y, \in \mathbb R $ we have that $f(x)\le f(y) \iff y \le x$