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I am reading the Wiki article about singularities and I was wondering what kind of singularity is the Dirac delta function not defined as a distribution but as this way:

\begin{equation} \delta^{\epsilon}(x) = \left\{ \begin{array}{rcl} \dfrac{1}{\epsilon}& \hspace{30pt}\text{if} \hspace{10pt}& -\dfrac{\epsilon}{2} <x<\dfrac{\epsilon}{2} \\ \\ 0 &\hspace{30pt} \text{if} \hspace{10pt}& |x|>\dfrac{\epsilon}{2} \end{array} \right. \end{equation}

With $\epsilon \rightarrow 0$. I would say that is an infinite discontinuity but I'm not sure.

Supposing that is an infinite discontinuity, how can I differentiate it with the singularity produced in $x=0$ by a function like:

$$ f(x)=1/x $$

Question: What is the main difference between these two singularities?

Jackozee Hakkiuz
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LongJohn
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  • for one, $\delta$ is not actually a function (it's what is called a distribution)! where as $f(x)=\frac{1}{x}$ is – Nick Castillo May 08 '21 at 11:34
  • A big difference is that $\delta^{\epsilon}(x)$ is integrable across $0$ and so the suggestion is that $\delta(x)$ is too and gives a step function, while this might not be the case with $f(x)=\frac1x$ – Henry May 08 '21 at 11:42
  • Did you mean $-\frac{\epsilon}{2}<x<\frac{\epsilon}{2}$ versus $\frac{\epsilon}{2}<x<\frac{\epsilon}{2}$? If so, this is the boxcar function (see https://en.wikipedia.org/wiki/Boxcar_function) and a slight variation on the rectangle function (see https://en.wikipedia.org/wiki/Rectangular_function). – Steven Clark May 08 '21 at 14:24
  • Yes, I mean $-\frac{\epsilon}{2}<x<\frac{\epsilon}{2}$. Edited! But in this case $\epsilon \rightarrow 0$ – LongJohn May 08 '21 at 14:29
  • That wikipedia article is about singularities of functions. Not about weird non-functions like this. – GEdgar May 08 '21 at 14:37
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    The $\delta$ distribution is not a function, and the family of functions $\delta^\epsilon$ you defined doesn't converge to anything when $\epsilon\to 0$. That is, when you do $\epsilon\to 0$ you don't get a function. In both cases, you cannot apply the concepts like "infinite discontinuity" because that is defined for functions only. Do you mean to analyse these objects as distributions? In that case you should talk about singularities of distributions, but I don't know if there is a classification for these. – Jackozee Hakkiuz May 08 '21 at 16:53
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    @JackozeeHakkiuz. On any compact subset every distribution is a derivative of some order of a continuous function. For example, $\operatorname{pv}\frac1x$ is the second derivative of $x(\ln|x|-1).$ This can probably be used for classification. – md2perpe May 09 '21 at 12:23
  • There are ways to measure how singular a distribution at a point. First there is the wave front set which tells you in what directions the distribution is singular https://ncatlab.org/nlab/show/wavefront+set There is also the Steinmann scaling degree https://ncatlab.org/nlab/show/scaling+degree+of+a+distribution – Abdelmalek Abdesselam May 10 '21 at 15:02
  • See the primer on distributions, including the Dirac Delta and its derivative, in THIS ANSWER. – Mark Viola May 10 '21 at 21:38

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