Bump Function Analysis Homework

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Bump Function

Given any open set in with compact closure, there exist smooth functions which are identically one on and vanish arbitrarily close to . One way to express this more precisely is that for any open set containing , there is a smooth function such that

1. for all and

2. for all .

A function that satisfies (1) and (2) is called a bump function. If then by rescaling , namely , one gets a sequence of smooth functions which converges to the delta function, providing that is a neighborhood of 0.

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