Null Function/Examples/Example 3

Example of Null Function

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

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

Then $f$ is not a null function.

Let $x > 2$.

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

$=$

$\ds \int_0^1 \map f u \rd u + \int_1^2 \map f u \rd u + \int_2^x \map f u \rd u$

$\ds $

$=$

$\ds \int_0^1 0 \rd u + \int_1^2 1 \rd u + \int_2^x 0 \rd u$

Definition of $\map f x$

$\ds $

$=$

$\ds 0 + 1 + 0$

Hence the result by definition of null function.

$\blacksquare$

1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Solved Problems: Impulse Functions. The Dirac Delta Function: $44 \ \text{(b)}$