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Often in physics we integrate by parts

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by:

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I have a really simple question, how can we assume that $[f(x)\delta (x-y){]}_{{x}_{0}}^{{x}_{1}}=0$?

Intuitively the delta function is zero except for at $x=y$, but what if either ${x}_{0}$ or ${x}_{1}$ was equal to y?

Is the answer simply 'we must assume separately that ${x}_{0},{x}_{1}\ne y$, or is there something obvious that I'm missing, or is there some measure theory reason why we can say it is zero?

$$

by:

$$

I have a really simple question, how can we assume that $[f(x)\delta (x-y){]}_{{x}_{0}}^{{x}_{1}}=0$?

Intuitively the delta function is zero except for at $x=y$, but what if either ${x}_{0}$ or ${x}_{1}$ was equal to y?

Is the answer simply 'we must assume separately that ${x}_{0},{x}_{1}\ne y$, or is there something obvious that I'm missing, or is there some measure theory reason why we can say it is zero?