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suppose $ f(x)\in C_c^{\infty}(R^n, R)$, an infinitely differentiable function with compact support, from $R^n$ to $R$.

If $f\not\equiv 0$, is the boundary of its support, i.e. $\partial\{x\in R^n: f(x)\neq 0\}$, always a smooth surface of $n-1$-dimension in $R^n$? Or at least piece-wise smooth? (of course it may comprise of disjoint components)

Thanks a lot :-D

Calvin
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1 Answers1

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Well you can definitely get a boundary which is only piecewise smooth....

Take $f \in C_c^\infty(R,R)$ whose support equals $[0,1]$. What is the support of $g \in C_c^\infty(R^2,R)$ given by $g(x,y) = f(x)f(y)$?

Mike F
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