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Suppose I have two functions that are Schwartz class, say $f,g \in S(\mathbb{R})$, and suppose I have another function $\psi(x)$ such that \begin{equation} g(x) = \psi(x)f(x) \end{equation} I would like to find a way to understand why this means that $\psi$ must be differentiable but I struggle to find a start as I am very new to Schwartz class functions.

I thought maybe smoothness is enough to begin with. In any case, away from points where $f(x) \neq 0$ I have no problems, but somehow I need to find a way to say something about the differentiability fo $\psi$ in general, provided $f,g$ are Schwartz (and not both identically zero).

Any help would be great !

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    ψ(x) must not only be differantiable but also mustn't grow faster than polynoms. This question might help you a little. http://math.stackexchange.com/questions/104565/if-fx-hxgx-is-h-differentiable-if-f-and-g-are2012-02-01

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