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I have been struggling to show that there does not exist a surjective continuous map from $\mathbb{R}^2$ into $\mathbb{R}$ such that the inverse image of any compact set is compact.

I know that such a function would map compact sets to compact sets. Are there any suggestions for solving this problem?

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Hint: assume $K=\{x:f(x)=0\}$ is compact. Take $R>0$ such that $K\subseteq B_R=\{x:\|x\|\le R\}$. Then $B_R$ is also compact and so $f[B_R]$ is bounded, say contained in $[-N,N]$ for some~$N$. Take $a$ and $b$ with $f(a)=-N-1$ and $f(b)=N+1$. Take a curve $C$ outside $B_R$ that connects $a$ and $b$ and show that there is a point on $C$ that maps to $0$.

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