What can be $f$ so that $f^2(x) = -x$ for all $x\in R$?
I know that if $f^2(x) = -x$ then $f(x)$ is injective and $f$ can not be continuous.
But I can not find an example of discontinuous function so that $f^2(x) = -x$ for all $x\in R$.
Can anyone help me?
Just look at the cycle representation of $f\circ f=-x$, there is one fixed element and a bunch of $2$-cycles, So to build a suitable permutation just pair up the $2$-cycles and for each $(a,b)$ and $(c,d)$ add the $4$-cycle $(a,c,b,d)$.