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$$ \varphi(x)=e^{1/(x^2-1)} $$ 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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    $\begingroup$ 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. $\endgroup$ – t.b. May 23 '11 at 21:34
  • $\begingroup$ 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. $\endgroup$ – Jonas Teuwen May 23 '11 at 23:41
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Bump function.

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  • $\begingroup$ 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 $e^{1/(x^2-1)}$. $\endgroup$ – leftaroundabout Sep 27 '11 at 21:10

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