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?
Are test/bump functions always bounded?
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functional-analysis
distribution-theory
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0This is an important property, but it's understood that continuous functions with compact support are bounded. – 2012-06-22
1 Answers
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Hint: The image of a compact set under a continuous function is always compact.