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Let $f : \mathbb{R} → \mathbb{R}$ be continuous, with $f(x)f(f(x)) = 1$ for all $x ∈ \mathbb{R}$. If $f(1000) = 999$, find $f(500)$.

I try to solve this problem, but I don't know how to use de continuity on $f$. Is anyone could give me a little hint on the continuity? And please, don't give me the answer to the question.

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Hint 1: What can you say about $f(y)$ if $y$ is in the image of $f$?

Hint 2: The intermediate value theorem will be helpful. Note that $999$ is in the image of $f$.

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  • $\begingroup$ I suspected that I could use the intermediate value theorem, but I do not know how. Thank you, I'll think now. $\endgroup$ – user230283 Oct 4 '15 at 17:08
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Hint Use Intermediate Value Theorem between $\frac{1}{999}$ and $999$.

$500$ belongs in this interval.

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