Existence of Bump Functions
It is possible to construct bump functions "to specifications". Stated formally, if is an arbitrary compact set in n dimensions and is an open set containing there exists a bump function which is 1 on and 0 outside of Since can be taken to be a very small neighborhood of this amounts to being able to construct a function that is 1 on and falls off rapidly to 0 outside of while still being smooth.
The construction proceeds as follows. One considers a compact neighborhood of contained in so The characteristic function of will be equal to 1 on and outside of so in particular, it will be 1 on and outside of This function is not smooth however. The key idea is to smooth a bit, by taking the convolution of with a mollifier. The latter is just a bump function with a very small support and whose integral is 1. Such a mollifier can be obtained, for example, by taking the bump function from the previous section and performing appropriate scalings.
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