# Definition:Null Function

## Definition

Let $\mathcal N: \R \to \R$ be a real function such that:

$\forall x \in \R_{>0}: \displaystyle \int_0^x \map {\mathcal N} t \rd t = 0$

Then $\mathcal N$ is a null function.

## Examples

### Example 1

Let $f: \R \to \R$ be the real function defined as:

$\forall x \in \R: \map f x = \begin {cases} 1 & : t = 1/2 \\ -1 & : t = 1 \\ 0 & : \text {otherwise} \end {cases}$

Then $f$ is a null function.

### Example 2

Let $f: \R \to \R$ be the real function defined as:

$\forall x \in \R: \map f x = \begin {cases} 1 & : t = 1 \\ 0 & : \text {otherwise} \end {cases}$

Then $f$ is a null function.

### Example 3

Let $f: \R \to \R$ be the real function defined as:

$\forall x \in \R: \map f x = \begin {cases} 1 & : t = 1 \le t \le 2 \\ 0 & : \text {otherwise} \end {cases}$

Then $f$ is not a null function.

## Also see

• Results about null functions can be found here.