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?