Bump function

In mathematics, a bump function is a function f : Rⁿ → R on a Euclidean space Rⁿ which is both smooth (in the sense of having continuous derivatives of all orders) and compactly supported. The space of all bump functions on Rⁿ is denoted or . The dual space of this space endowed with a suitable topology is the space of distributions.

Examples

The function Ψ : R → R given by

is an example of a bump function in one dimension. It is clear from the construction that this function has compact support, since a function of the real line has compact support if and only if it has bounded support. The proof of smoothness follows along the same lines as for the related function discussed in the Non-analytic smooth function article. This function can be interpreted as the Gaussian function scaled to fit into the unit disc: the substitution corresponds to sending x = ±1 to y = ∞.

A simple example of a bump function in n variables is obtained by taking the product of n copies of the above bump function in one variable, so