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For each closed set $A\subseteq\mathbb{R}$, is it possible to construct a real continuous function $f$ such that the zero set, $f^{-1}(0)$, of $f$ is precisely $A$, and $f$ is infinitely differentiable on all of $\mathbb{R}$?

Thanks!

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    Corresponding question on MathOverflow: http://mathoverflow.net/questions/240342012-01-12

3 Answers 3