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Does there exist a continuous function $f :\mathbb{R} \rightarrow \mathbb{R}$ such that $f \circ f(x) = -x$ for all $x$?

My feeling is that there isn't, but I don't know how to go about proving this.

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Since $f$ is bijective, as $f^{-1} = -f$, and is continuous, $f$ needs to be increasing or decreasing. But the composition of increasing functions is increasing. And composition of decreasing functions is increasing.

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