Are test/bump functions always bounded? 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?

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