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When I was doing revision for definite integral, I got confused with one step.

Here is the statement:

For all continuous functions $f$ which are continuous at $0$:

$$\lim_{ε\to 0}\int_{-ε}^ε f(x)·D(x,ε)dx = f(0)·\lim_{ε\to 0}\int_{-ε}^ε D(x,ε)dx$$

where $D(x,ε)$is the Dirac Delta function.

But I don't really understand why is this true. Could anyone tell me the steps between them?


Rewritten as a teaching exercise:

Please explain why for all functions $f$ continuous at $0$:

$$\lim\limits_{\epsilon \to 0} \int\limits_{-\epsilon}^\epsilon f(x) \delta(x)\ dx = f(0) \lim_{\epsilon \to 0}\int\limits_{-\epsilon}^\epsilon \delta(x)\ dx = f(0).$$

where $\delta$ is a Dirac delta function.


Here is the definition of Dirac delta function

$$\lim_{\epsilon \to 0}\delta(t;\epsilon)=0\quad\forall t\neq0$$ and $$\lim_{\epsilon \to 0}\int_{-\infty}^\infty \delta(t;\epsilon)dt=1$$

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  • $\begingroup$ Take $f=1$ a constant function. Then the integral equals $2\epsilon$ which tends to $0$ when $\epsilon \rightarrow 0$. $\endgroup$ Apr 19, 2020 at 19:24
  • $\begingroup$ @ANOOB: You're new here (as revealed by your reputation), so I urge you to learn from experts here on how to ask question. "When I was doing revision for definite integral" is both ungrammatical ("doing revision"??), but also unnecessary and hence a waste of your readers' time. Also, the fact that you're "confused" is implicit in the fact that you're asking a question. You don't need to restate it. You also have many errors in MathJax, which is an added burden on readers. "Here is the statement" is redundant: just give the statement. I'll post a better question below, so you see. $\endgroup$ Apr 19, 2020 at 19:49
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    $\begingroup$ What is your definition of $D(x,\varepsilon)$ ? $\endgroup$
    – Tuvasbien
    Apr 19, 2020 at 19:50
  • $\begingroup$ @DavidG.Stork Thank you so much for the correction! This is my first time asking a question so sorry about the errors I have made! $\endgroup$
    – ANOOB
    Apr 20, 2020 at 5:12
  • $\begingroup$ @Tuvasbien I have added the definition, thank you! $\endgroup$
    – ANOOB
    Apr 20, 2020 at 5:28

1 Answer 1

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Consider the function $f(x)=6$. Then, as you can see easily the integral $\int_{-\epsilon}^{\epsilon}f(x)=0$ , as $\epsilon$ goes to 0, not $f(0)$.

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