TITLE: Finding functions satisfying $f(f(x))+f(x)+x=0$
QUESTION [9 upvotes]: Find the number of continous functions $f:\Bbb R\to\Bbb R$ which satisfy the following functional equation $$f(f(x))+f(x)+x=0$$

REPLY [4 votes]: Another proof when $f(x)$ is continuous.
As already noted by Michael Burr and Eric Wofsey, $f$ satisfied $f^{(3)}(x)=x$, which shows that $f$ is a bijection. So, $f$ is monotone as a continuous injection.
The fucntions $f$ and $f^{-1}$ share the same monotonicity, and since $f(x)+f^{-1}(x)=-x$, $f$ must be decreasing. Consequently, $f^{(3)}(x)$ is decreasing, which gives a contradiction since $f^{(3)}(x)=x$.