Distributional Derivative of Heaviside Step Function

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Theorem

Let $H : \R \to \closedint 0 1$ be the Heaviside step function.

Let $T \in \map {\DD'} \R$ be a distribution corresponding to $H$.

Let $\delta \in \map {\DD'} \R$ be the Dirac delta distribution.


Then the distributional derivative of $T$ is $\delta$.


Proof

Let $\phi \in \map \DD \R$ be a test function.

Then:

\(\ds \map {T'} \phi\) \(=\) \(\ds -\map T {\phi'}\) Definition of Distributional Derivative
\(\ds \) \(=\) \(\ds -\int_{-\infty}^\infty \map H x \map {\phi'} x \rd x\)
\(\ds \) \(=\) \(\ds -\int_0^\infty \map {\phi'} x \rd x\)
\(\ds \) \(=\) \(\ds \map \phi 0 - \map \phi \infty\)
\(\ds \) \(=\) \(\ds \map \phi 0\) Definition of Test Function, Value of Compactly Supported Function outside its Support, $\infty$ is not in a bounded subset of real numbers
\(\ds \) \(=\) \(\ds \map \delta \phi\) Definition of Dirac Delta Distribution

$\blacksquare$


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