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I can not understand this mathematical formula:

$$\int_{a-\epsilon}^{a+\epsilon}f(x)\delta(x-a)dx=f(a)$$

I understand that it is the derivative of an integral evaluated in $a$, but still can not explain in mathematically.

How do you get $f(a)$ from the integral side? and what are the assumptions you have to do?

Vincenzo Tibullo
  • 10,955
  • That is usually how the $\delta$ is defined, it is "something that if you do this integral together multiplied with any (sufficiently well behaving) $f$ the result becomes $f(a)$". – mathreadler Apr 29 '17 at 17:56
  • Yes @IsaacBrowne is right the definition has integral limits $-\infty,\infty$. But which function could give $f(a)$ if $\delta$ does any considerable things to $f$ far away from $a$. – mathreadler Apr 29 '17 at 17:59
  • There should be a mathematical demonstration of that equality –  Apr 29 '17 at 18:07
  • I think it is dependent which space of test functions $f$ belongs to. A popular choice is the Schwartz class of infinitely smoothly differentiable functions $f\in \mathcal C^\infty(\mathbb R)$ – mathreadler Apr 29 '17 at 18:09
  • Thanks. So in that case f(a) is out of the integral an the rest is simple.., but still i am not sure.. –  Apr 29 '17 at 18:11

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I find a great answer here (pags 2-3)