Lipschitz bump function

Question

Does anyone know of an example of a Lipschitz or Holder continuous bump function on $\mathbb{R}^n$? Any help is appreciated. Thank you.

Bump functions typically are smooth and have compact support, hence they are Lipschitz. — copper.hat, Dec 04 '13 at 19:02
I suppose the smoothness and the boundedness of the gradient makes it Lipschitz? — canis89, Dec 04 '13 at 23:19
Sorry, I should have said that it is because of the mean value theorem. — copper.hat, Dec 05 '13 at 01:00
Yes, with the mean value theorem and the fact that the gradient is bounded, we can indeed imply that it is Lipschitz. — canis89, Dec 05 '13 at 01:01

score 1 · Accepted Answer · answered Dec 04 '13 at 19:02

1

The usual bump function is $C^\infty$ so has all the properties you want.

answered Dec 04 '13 at 19:02

Igor Rivin

actually, they aren´t analytical functions on $\mathbb{R}$, so caution is required. – Joako Feb 14 '22 at 18:33

1 Answers1