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A
-shaped sequence is a sequence of nonnegative smooth functions equal to outside neighborhoods that tend to as and each possessing an integral of . -
The Dirac Delta Function is defined such that
Of course, no such function exists, but we treat it as if it were any other mathematical object.
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For any continuous function
, we have And also
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Translation by
along the -axis preserves this property We refer to this as the
- function concentrated on denoted .