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a bump function is a infinitely often differentiable function with compact support. I guess that such functions are always bounded, especially because the set where they are not zero is compact and because they are continuous they should attain a maximum value on that set. or am i wrong? i am wondering because nowhere in the literature i am using there it is said that such functions are bounded, and i guess this is an important property and think it should be mentioned if it holds. so maybe its not the case?

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They are bounded. So are their derivatives. It's one of things that are considered too obvious to mention. – user31373 Jun 19 '12 at 20:39
ok, so i was right, thx :) – Stefan Jun 19 '12 at 20:42
You should check out the extreme value theorem. – N.U. Jun 19 '12 at 20:45
This is an important property, but it's understood that continuous functions with compact support are bounded. – Qiaochu Yuan Jun 22 '12 at 14:10
up vote 2 down vote accepted

Hint: The image of a compact set under a continuous function is always compact.

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