Decay of smooth functions around their boundary Code Answer

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If $fin{cal C}^infty({bf R}^n)$ and $(x_k)_k$ is a converging sequence with $d(x_k,{rm supp}(f)^C)le 1/k$, does $k^N f(x_k)$ converge to $0$ then for any fixed $N>0$?

Answer

Assume the sequence converges to $x_0in partial text{supp} (f)$. Since all derivatives exist and are bounded in a neighborhood of $x_0$, the Taylor expansion with remainder tells us that, for a fixed $a>0$ and $N>0$, $f(x)=o(|x-y|^N)$ for every $xin (x_0-a, x_0+a)$ and $yin partial text{supp}(f)$. Now set $x=x_k$ and $y=y_k$ such that $|x_k-y_k|=text{dist}(x_k, text{supp}(f)^c)$. This is your claim.

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