# Definition:Test Function

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

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.

## Sources

- 2017: Amol Sasane:
*A Friendly Approach to Functional Analysis*... (next): Chapter $\S 6.1$: A glimpse of distribution theory. Test functions, distributions, and examples