In mathematical analysis, a bump function (also called a test function) is a function ${\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }$ on a Euclidean space ${\displaystyle \mathbb {R} ^{n}}$ 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 ${\displaystyle \mathbb {R} ^{n}}$ forms a vector space, denoted ${\displaystyle \mathrm {C} _{0}^{\infty }(\mathbb {R} ^{n})}$ or ${\displaystyle \mathrm {C} _{\mathrm {c} }^{\infty }(\mathbb {R} ^{n}).}$ The dual space of this space endowed with a suitable topology is the space of distributions.

The function ${\displaystyle \Psi :\mathbb {R} \to \mathbb {R} }$ given by $${\displaystyle \Psi (x)={\begin{cases}\exp \left({\frac {-1}{1-x^{2}}}\right),&{\text{ if }}|x|<1,\\0,&{\text{ if }}|x|\geq 1,\end{cases}}}$$

is an example of a bump function in one dimension. Note that the support of this function is the closed interval ${\displaystyle [-1,1]}$. In fact, by definition of support, we have that ${\displaystyle \operatorname {supp} (\Psi ):={\overline {\{x\in \mathbb {R} :\Psi (x)\neq 0\}}}={\overline {(-1,1)}}}$, where the closure is taken with respect the Euclidean topology of the real line. 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 ${\displaystyle \exp \left(-y^{2}\right)}$ scaled to fit into the unit disc: the substitution ${\displaystyle y^{2}={1}/{\left(1-x^{2}\right)}}$ corresponds to sending ${\displaystyle x=\pm 1}$ to ${\displaystyle y=\infty .}$

A simple example of a (square) bump function in ${\displaystyle n}$ variables is obtained by taking the product of ${\displaystyle n}$ copies of the above bump function in one variable, so $${\displaystyle \Phi (x_{1},x_{2},\dots ,x_{n})=\Psi (x_{1})\Psi (x_{2})\cdots \Psi (x_{n}).}$$

A radially symmetric bump function in ${\displaystyle n}$ variables can be formed by taking the function ${\displaystyle \Psi _{n}:\mathbb {R} ^{n}\to \mathbb {R} }$ defined by ${\displaystyle \Psi _{n}(\mathbf {x} )=\Psi (|\mathbf {x} |)}$. This function is supported on the unit ball centered at the origin.

For another example, take an ${\displaystyle h}$ that is positive on ${\displaystyle (c,d)}$ and zero elsewhere, for example

Smooth transition functions

Consider the function