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Bump Function


BumpFunctionCurve

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

1. f(x)=1 for all x in U and

2. f(x)=0 for all x not in V.

Bump function

A function f that satisfies (1) and (2) is called a bump function. If intf=1 then by rescaling f, namely f_k(x)=k^nf(kx), one gets a sequence of smooth functions which converges to the delta function, providing that U is a neighborhood of 0.


See also

Compact Support, Convolution, Delta Function, Smooth Function

This entry contributed by Todd Rowland

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Rowland, Todd. "Bump Function." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/BumpFunction.html

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