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This question already has an answer here:

Consider a functions $f:\mathbb R\to \mathbb R$ such that $f(f(x))=-x$ for all $x$. It is shown in

``f(f(x)) = − x, Windmills, and Beyond" by Martin Griffiths which appeared in Mathematics Magazine, Vol. 83, No. 1 (February 2010), pp. 15-23

that the set of discontinuities say $D_f$, must be at least countable. Can one obtain a good description of $D_f$? Can it be a Cantor set for example? It is known that $D_f$ is an $F_{\sigma}$ set. If an $F_{\sigma}$ set is symmetric with respect to the origin and contains the origin, can it be equal to $D_f$ for such a ``windmill" function $f$?

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marked as duplicate by Jyrki Lahtonen, user416281, Masacroso, Dietrich Burde, RRL real-analysis Dec 3 '18 at 19:56

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

  • $\begingroup$ I removed the tag "descriptive set theory" because the question doesn't seem to have anything to do with that subject. If I'm wrong, please forgive me, and explain the relation when you put the tag back. $\endgroup$ – saulspatz Dec 3 '18 at 18:17
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    $\begingroup$ The idea is that $f$ must permute real numbers (other than zero) in 4-cycles in such a way that two pairs of negatives are included in each cycle. See this thread. $\endgroup$ – Jyrki Lahtonen Dec 3 '18 at 18:25
  • $\begingroup$ Anyway, one of the answers explains why such a function must have an infinite number of discontinuities. I think the example function I gave is measurable (if that is of interest to you). $\endgroup$ – Jyrki Lahtonen Dec 3 '18 at 18:28
  • $\begingroup$ Anyway, this is very close to being a duplicate of that. I will vote to close it as such. If you want to ask about some other property of such a function, edit that into the question. Also @-ping me, and I will reconsider my vote. There may be some scope for further questions about properties of such functions. $\endgroup$ – Jyrki Lahtonen Dec 3 '18 at 18:30