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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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replace $e^x$ with something else that grows very quickly – yoyo Mar 31 '11 at 20:21

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You can convolute a characteristic function with a mollifier (however, a mollifier can ultimately be some form of an exponential function).

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I was thinking along the same lines: you can multiply any function you like by a $\psi_i$ from a partition of unity w.r.t. an open cover of your choice, but then again, the existence of the latter is often proved using the much-loved exponential function. – Gerben Mar 31 '11 at 22:23

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