This question is related, but more restrictive than the one asked here. Namely, suppose you have a $C^k$ function on $[a,b]$. Can you extend it to a function in $C_c^k(\mathbb{R})$, i.e. a function of compact support on $\mathbb{R}$ ? The extension proposed in the other question doesn't seem to satisfy this condition of compact support.
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In the trivial case of a function identically $1$ on $[a,b]$ it's ok, because we know there exists bump functions, i.e. smooth functions taking value $1$ on $[a,b]$ and zero say outside $[a-1,b+1]$. But for general functions in $C^k[a,b]$ ? – Lonewolf Aug 14 '20 at 15:29
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2What happens if you multiply an arbitrary $C^k$ extension with a bump function? – Daniel Fischer Aug 14 '20 at 15:53
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Well, what happens is that I realize I'm a bit stupid.. Thank you ! – Lonewolf Aug 14 '20 at 20:06