Determine all continuous real function which satisfies the following

Question

We are required to determine all continuous real valued functions $f$ such that $$f(f(x))=-x$$

I’ve determined that if such a function exists, it must be bijective. But I don’t know if such a function exists, let alone find all such functions. Any hints and suggestions will be appreciated.

Answer 1

Since it is injective, continuous and its domain is an interval, it must be monotonic. Then, $f\circ f$ is an increasing functions. Therefore, no such function exists.