Definition:Null Function

Definition

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

$\forall x \in \R_{>0}: \ds \int_0^x \map \NN t \rd t = 0$

Then $\NN$ 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

• Definition:Null Measure

• Results about null functions can be found here.

Sources

• 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Some Special Functions: $\text {IX}$. Null functions