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 it's not the case?
2026-04-08 19:34:33.1775676873
Are test/bump functions always bounded?
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Hint: The image of a compact set under a continuous function is always compact.