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I would like to give you the physics context in which this question emerged, but that would be a very long explanation (sorry!). So, since the question is almost self contained, I am just going to tell you what it is.

Consider the product Y(x)=H(x)(1-H(x)), where H(x) is the Heaviside step function. Of course, Y(x) is zero everywhere. Except, at the origin there is a big problem: the value of Y(0)=H(0)(1-H(0)) depends on the choice of H(0).

To make things worse, in the context where this question emerged, there is no particular reason to choose a specific value for H(0).

To make things even worse, apparently one cannot use the theory of distributions, since the product of distributions is ill-defined!

Does anyone have any suggestions for a rigorous way to deal with the problem?

Thanks a lot!

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# Product of opposite Heaviside Steps

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