Help Understanding Dirac Delta Function in Lecture Notes

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The Dirac delta function serves as the continuous counterpart to the discrete Kronecker delta function, acting as a mathematical tool to represent point sources or impulses. It is defined such that it is zero everywhere except at a single point, where it is infinitely high, ensuring that its integral over the entire space equals one. Understanding its properties is crucial for applications in physics and engineering, particularly in signal processing and differential equations. The linked Khan Academy video provides a clear explanation of its significance and applications. Mastery of the Dirac delta function is essential for grasping advanced concepts in calculus and related fields.
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I don't understand in the first paragraph of the attached lecture notes.

Could anyone help?
 

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To begin with, why is the Dirac delta function is the continuous equivalent of the discrete-case Kronecker delta function.
 

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