# Definition:Test Function

## Definition

Let $\phi : \R^d \to \C$ be a complex-valued function.

Let $\phi$ be a smooth function across its whole domain (including boundaries of support) with respect to all its variables.

Let $K \subseteq \R^d$ be a compact subset.

Suppose $\phi$ vanishes outside of $K$:

$\forall x \in \R^d \setminus K : \map \phi x = 0$

Then $\phi$ is known as a test function.

## Also known as

Test functions are also known as bump functions.

## Examples

### Exponential of $\dfrac 1 {x^2 - 1}$

The graph of the test function. It is smooth everywhere. Outside of its support denoted by dashed lines the function is identically zero. At boundary points it connects smoothly.

Let $\phi : \R \to \R$ be a real function with support on $x \in \closedint {-1} 1$ such that:

$\map \phi x = \begin {cases} \map \exp {\dfrac 1 {x^2 - 1} } & : \size x < 1 \\ 0 & : \size x \ge 1 \end {cases}$

Then $\phi$ is a test function.

## Also see

• Results about test functions can be found here.