## Examples of smooth functions with compact support [closed]

Are there functions in $C_0^\infty(\mathbb R)$ which are not related to $f(x)=\begin{cases}e^{-1/x}&x>0\newline 0&x\leq 0\end{cases}$ by translation, dilation, or composition?

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 check "convolution" in any textbook of fun. anal. or in the wiki article – Pietro Majer Apr 1 2011 at 5:36