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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.


marked as duplicate by Jyrki Lahtonen, José Carlos Santos real-analysis Oct 28 '18 at 16:42

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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.

  • $\begingroup$ Ah, nice. How did I miss that! Thanks for taking your time. $\endgroup$ – John Mitchell Oct 28 '18 at 16:38

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