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 $f$?

  • $\begingroup$ 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, 2018 at 18:17
  • 1
    $\begingroup$ 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. $\endgroup$ Dec 3, 2018 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). $\endgroup$ Dec 3, 2018 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. $\endgroup$ Dec 3, 2018 at 18:30


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