# Bump function

## Smooth and compactly supported function / From Wikipedia, the free encyclopedia

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In mathematics, a **bump function** (also called a **test function**) is a function $f:\mathbb {R} ^{n}\to \mathbb {R}$ on a Euclidean space $\mathbb {R} ^{n}$ which is both smooth (in the sense of having continuous derivatives of all orders) and compactly supported. The set of all bump functions with domain $\mathbb {R} ^{n}$ forms a vector space, denoted $\mathrm {C} _{0}^{\infty }(\mathbb {R} ^{n})$ or $\mathrm {C} _{\mathrm {c} }^{\infty }(\mathbb {R} ^{n}).$ The dual space of this space endowed with a suitable topology is the space of distributions.