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Let $\varOmega \subset\mathbb R^n$ be a bounded connected open set and $r$ a positive integer.

Given $f\in C^r (\overline \varOmega)$ (i.e. $f$ is $C^r$ on $\varOmega$ and it extends continuously, together with all its partial derivatives up to order $r$, to the whole $\overline \varOmega$).

Question: Does it follows, without further assumption on $\varOmega$, that $f$ has a $C^r$ extension to the whole $\mathbb R^n$?

The answer is negative, as pointed out by Willie Wong - see the counterexample in

Density of polynomials in $C^k(\overline\Omega)$

The answer is positive if $\varOmega$ is Lipschitz.

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    $\begingroup$ Your question is answered in the negative as part of the question statement of this question. $\endgroup$
    – Willie Wong
    Commented Oct 26, 2016 at 13:07
  • $\begingroup$ Are you familiar with Whitney extension theorem? I think it's answer this question completely. $\endgroup$
    – Oleg Eroshkin
    Commented Oct 26, 2016 at 15:58
  • $\begingroup$ Willie Wong, very elegant counterexample, thank you for posting the link. $\endgroup$
    – Nautilus
    Commented Oct 28, 2016 at 14:43

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