# What is the name for the archetypical example of a test function, $\varphi(x)=e^{1/(x^2-1)}$?

$$\varphi(x)=e^{1/(x^2-1)}$$ This function (on the interval $\quad]\!-1,1[ \,\,\,$, outside of it simply $\equiv0$) is used as the typical example of a test function / bump function, I have so far seen it it every book that covers $\mathcal{C}_0^\infty$ functions. But it's usually not called any specific name, though it does seem to have one, at least I heard it being called by some name recently, but forgot it.

I'd greatly like to know a name for this function, both for my computer functions library and for ease when writing proofs where a test function is required, and you can quickly reassure its existence with a simple "like the ...-function".

Friedrichs'sche Glättungsfunktion is in fact the name I was looking for!

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In German it is sometimes called Friedrichs'sche Glättungsfunktion (roughly: Friedrichs's mollifying function) to honour its use in Friedrichs's work on differential equations. You can find a discussion and references on the Wikipedia-page on mollifiers. I don't know how "official" that name is, however. – t.b. May 23 '11 at 21:34
Usually you don't need an explicit formula for your bump function right? You just need to know that it exists. This functions allows you to prove that such $C^\infty$ bump functions exist. – Jonas Teuwen May 23 '11 at 23:41

## 1 Answer

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Ahm... it would be a bit unfair to accept this answer now, considering Theo already gave the desired answer in the comments and bumb function refers to general functions of this type, not specifically $e^{1/(x^2-1)}$. – leftaroundabout Sep 27 '11 at 21:10
Your problem, not mine. In fact, I could not care less... – Did Sep 27 '11 at 21:14
This is interesting in the sense that I hadn't considered that html links would contribute to message length. Hmm. – mixedmath♦ Sep 29 '11 at 22:00
@DidierPiau : Yes it is his problem, It has happened to me too! To choose between 2 correct answers is never fair. Why multiple ansers are not allowed? after all both answers are correct. – Arjang Nov 26 '11 at 13:44

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The other is not an answer. – timur Dec 3 '11 at 19:47
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