Bump function

In mathematical analysis, a bump function is a localized auxiliary function, usually chosen to be smooth and to have compact support. Bump functions are commonly used as cutoff functions, for example functions that are equal to 1 on a prescribed set and vanish outside a larger set, and as standard examples of kernels used to construct mollifiers.

Some authors use the term more broadly for any compactly supported smooth function. Such functions are important examples of test functions, especially in distribution theory, but the terms "bump function" and "test function" are not synonymous in all contexts.