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I am supposed to give a counterexample showing the conclusion is false when $A$ is not closed. I tried to find one when $M$ is Euclidean space but kept failing... Could anyone please show me a counterexample?

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  • $\begingroup$ If $A$ is not closed there might not be continuous extension (think of $A$ as an interval minus a point) $\endgroup$ – user99914 Jan 19 '18 at 17:29
  • $\begingroup$ I can think of sets but I cannot find a specific smooth function on $A$. Could you give me more hints? $\endgroup$ – Keith Jan 19 '18 at 17:32
  • $\begingroup$ If it can be extended, the left and right limit has to agree. $\endgroup$ – user99914 Jan 19 '18 at 17:33
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Consider $f(x)=1/x$ defined on $\mathbb{R}-\{0\}$, you cannot extend it at $0$

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  • $\begingroup$ how to show that it cannot be extend to $0$ $\endgroup$ – Uncool Apr 29 at 13:42

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