Function with the property $f\circ f=-id$ [duplicate]

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 $$f$$?

marked as duplicate by Jyrki Lahtonen, user416281, Masacroso, Dietrich Burde, RRL real-analysis StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Dec 3 '18 at 19:56

• 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. – Jyrki Lahtonen Dec 3 '18 at 18:25