# 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.

## marked as duplicate by Jyrki Lahtonen, José Carlos Santos 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(); } ); }); }); Oct 28 '18 at 16:42

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.