## Differentiable function germs on differentiable manifolds

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Hello everyone, I was wondering if anyone knew how to prove that the map from $C^{\infty}(M)$ to $\xi (p)$, that is, from the infinitely differentiable functions on a manifold M to the space of (once)-differentiable function germs, where the map is associating to each f in $C^{\infty}(M)$ its class in $\xi (p)$ is onto. By the way, since you ask, the reason I'm interested in this is because its a question that WAS on my final for differential topology, I've tried to work it out since then but no luck so far, this is not homework it's just curiosity now, hope its ok ill have to check the post regultaions, sorry, if not just tell me and i'll delete the question...

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1. Your question's well-formed, but nearly impossible to parse. Edit? 2. I don't know any differential geometry, but is this homework? It reads kind of like it might be, and it's gotten downvoted. If it's homework, don't bother editing. – Harrison Brown Dec 6 2009 at 0:37

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Assuming you mean $C^{\infty}$ germs, then this is basically the existence of a cutoff function. – Akhil Mathew Dec 6 2009 at 2:12
Upvoted since reasonableness should be encouraged. :) – Harrison Brown Dec 6 2009 at 2:21

## 1 Answer

I think your question is mis-stated, because this map is not onto:

There is no smooth function on the reals whose germ at $0$ is the germ of $x|x|$, a once-differentiable function germ.

Assuming this is not what you wanted, please restate the question, and include an explanation of why you are interested in it ;)

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Thanks, I'll look into your counterexample! – unknown (yahoo) Dec 6 2009 at 1:17
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