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Suppose we've a sequence $f_{n}(x)$ of $C^{\infty}$ functions, with compact support in $R^{n}$. Define $h_{n}(x) = \inf\{f_{n}(x), f_{n+1}(x),....\}$. Is this $h_{n}(x) \in C^{\infty}$? Thanks

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The question makes no sense as it stands, as the functions are defined on different domains $\mathbb R^n$. Presumably this was meant to be $\mathbb R^k$ with a different variable $k$? – joriki Sep 29 '11 at 5:49

no it isn't in general. Consider for example a $C^\infty$-function $f\colon\mathbb R \to \mathbb R$, which is 0 outside $[-2,2]$ and equals $x\mapsto x$ in $[-1,1]$. Then $\inf\{f, -f\}$ equals $-|\,\cdot\,|$ in $[-1,1]$.

AB, martini.

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