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I'm trying to think of an example of a continuous function $f:\mathbb{R} \to \mathbb{R}$ such that $f^{-1}(X)$ is not compact but $ X \subset \mathbb{R}$ is compact. Any ideas?

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    @Arturo. Right, I missed that obvious point.2012-03-05

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What about the the constant function $f(x)=0$?

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Note that if $X$ is compact, it is closed, and so $f^{-1}(X)$ is closed. Now take your favourite set that is closed but not compact, call that $B$, and let $f(x) = \text{dist}(x,B)$. That is a continuous function on $\mathbb R$, and $B = f^{-1}(\{0\})$.

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Consider the continuous function $\sin x$