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Suppose you are given a $C^1$-function $f:\mathbb R^n\to\mathbb R$ which restricts to a smooth function $f|_{\partial B}:\partial B\to \mathbb R$, where $B$ is the unit ball in $\mathbb R^n$. Can one construct a sequence of $C^\infty$-functions $g_n:\mathbb R^n\to \mathbb R$ such that $g_n=f$ on $\partial B$ for every $n$ and such that $g_n$ converges in the $C^1$-norm to $f$ or is there an obvious reason why this can't be found? Thanks for any ideas.

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