I'm trying to generalize the property of the Kronecker delta function which gives(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\sum\nolimits_{i = 0}^n {{\delta _{ij}}} = \left\{ {\begin{array}{*{20}{c}}

1&{0 < j < n\,\,\,\,\,\,\,\,\,\,\,}\\

0&{j < 0\,\,or\,\,n < j}

\end{array}} \right\}\,\,.[/tex]

The continuous case seems to be the Dirac delta function such that

[tex]\int_R {{\rm{\delta (x - }}{{\rm{x}}_0}){\rm{dx}}} = \left\{ {\begin{array}{*{20}{c}}

{\begin{array}{*{20}{c}}

1&{{x_0} \in R}

\end{array}}\\

{\begin{array}{*{20}{c}}

0&{{x_0} \notin R}

\end{array}}

\end{array}} \right\}\,\,.[/tex]

But only using the Dirac delta function seems too restrictive for most applications. I'd like to keep the property of the integral being either 0 or 1, depending on whether some parameter,x, is or is not within the limits of the integral. But I can't think of any other function for which this is true. It seems that any other continuous function defined only in_{0}Rwill give the same integral no matter ifxis inside or outside_{0}R. So it seems the only way to insure thatxis always within_{0}Ris to makeRbe the whole real line from -∞ to +∞, in which case there is no integration to 0 sincexis always within_{0}R.

But I'm sure I don't know everything. And someone here might know something I don't.

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# A A function whose integral is either 0 or 1

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