When we truncate a function $f$ we usually consider $g_n(x)=f(x)\mathbb{1}_{[-n,n]}(x)+f(n)\mathbb{1}_{(n,\infty)}(x)+f(-n)\mathbb{1}_{(-\infty,-n)}(x)$, where $n$ is a natural number. Can we do it differently (in a standard, non case-to-case way) so that if $f\in C^k$, we get $g_n\in C^k$ such that $g_n=f$ if $x\in [-n,n]$ and $g_n$ is bounded outside $[-n,n]$?
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