Null Function/Examples/Example 3
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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.
Proof
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 $\map f x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 0 + 1 + 0\) | Definite Integral of Constant |
Hence the result by definition of null function.
$\blacksquare$
Sources
- 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: $44 \ \text{(b)}$