In texts about dirac delta,you often can find sentences like "The delta function is sometimes thought of as an infinitely high, infinitely thin spike at the origin".(adsbygoogle = window.adsbygoogle || []).push({});

If we take into account the important property of dirac delta:

[itex] \int_\mathbb{R} \delta(x) dx=1 [/itex]

and the fact that it is zero everywhere except at the origin,it seems that we can have an explanation for the quoted sentence.It can be said that the area under the curve of dirac delta should be one but it's width is zero so its height should be infinite.

Everything seems to work out well until it is said that,rigorously,dirac delta isn't a function becauseno function which is zero everywhere except at one point,can have non-zero definite integral.

But the explanation I gave for the dirac delta being infinite at origin,can be applied to a function like f(x) defined as [itex] f(0)=\infty \ \mbox{and} \ f(x \neq 0)=0 [/itex]

The definite integral of f(x) can be thought of as the area of a rectangle having zero width

and infinite height so having a finite and possibly non-zero area.

So it seems that's not a good reason for telling that dirac delta isn't a function.

I'll appreciate any idea.

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# A point about dirac delta

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