Does there exist a continuous function $f\colon \Bbb R\rightarrow \Bbb R$ such that $f(f(x))=-x$ for all $x\in\Bbb R$? Does there exist a continuous function

$f\colon \Bbb R\rightarrow \Bbb R$ such that $f(f(x))=-x$ for all $x\in\Bbb R$?


Hint: If $f(f(x))=-x$ then $f$ is a bijection and because $f$ is continuous it must also be either order preserving or order reversing.
