# Determine all continuous real function which satisfies the following [duplicate]

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.

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.