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This question is related to this post, for which I received a really good answer that gave a beautifull solution, yet I am still trying to understand one thing on the side that is not covered by the answer, which is the following claim:

Suppose we take a Schwartz function $f \in S(\mathbb{R})$ that satisfies $f(x_0) = 0$. Then, we have

\begin{equation} g(x) = (x - x_0)^{-1}f(x) \quad \in S(\mathbb{R}) \quad (x \neq x_0) \end{equation}

I am struggeling to come up with an idea to show this without reverting to the plain definition of a Schwartz function and trying to find the bounds on $g$. But is there another , more immediate answer? From my notes it sounds like there is, but there are no immediate steps.

The answer in the post that I linked this one to actually constructs g specifically, however in my notes the claim is made without any reference to a construction so I am guessing that I miss something here with regards to the properties of g that are imposed on it simply by the equation above... Thanks so much for your help!

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You want to define g(x_0) = f'(x_0) as well.

You might note that g(x) = \int_0^1 f'(x_0 + s(x-x_0)) \ ds; this makes it easy to show that $g$ is $C^\infty$ and satisfies the appropriate bounds.

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    okay, thanks a lot !!2012-01-31