Null Function/Examples/Example 1

Example of Null Function

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.

Let $x > 1$.

$\ds \int_0^x \map f u \rd u$

$=$

$\ds \int_0^{1/2} \map f u \rd u + \int_{1/2}^1 \map f u \rd u + \int_1^x \map f u \rd u$

$\ds $

$=$

$\ds \int_0^{1/2} 0 \rd u + \int_{1/2}^1 0 \rd u + \int_1^x 0 \rd u$

Definition of $\map f x$

$\ds $

$=$

$\ds 0 + 0 + 0$

The cases where $x < \dfrac 1 2$ and $\dfrac 1 2 \le x \le 1$ are treated similarly.

$\blacksquare$

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