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Possible Duplicate:
Periodic orbits

Suppose that $f$ is a continuous map from $\mathbb R$ to $\mathbb R$, which satisfies $f(f(x)) = x$ for each $x \in \mathbb{R}$.

Does $f$ necessarily have a fixed point?

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    Yes. Check this http://math.stackexchange.com/questions/233246/periodic-orbits/233252#2332522012-12-22

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