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Suppose I have a set of functions $(f_\epsilon)$ such that as $\epsilon\to 0$, $f_\epsilon\to F$ s.t.

$F(x)=0$ for $x\neq 0$ and $F(x)=\infty$ for $x=0$;

$\int_{-\infty}^\infty f_\epsilon(x) dx=1$ for all $\epsilon >0$

Then can I conclude that the limit is the delta function $\delta(x)$? (which has the sampling property too)

Thanks

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    You say $f_{\varepsilon}\to F$, but in that sense?2011-12-25
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    @DavideGiraudo: for example, say $f_\epsilon={\epsilon\over x^2 +\epsilon}$2011-12-25
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    How do you define "the delta function $\delta (x)$"?2011-12-25
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    @joriki: Pretty much it having the properties of $F$ PLUS the sampling property. I guess I am mainly interested in whether a function $F$ being the limit of functions $f_\epsilon$ would naturally have the sampling property. If it is not generally true, when would it be true, perhaps in the example I posted as a response to Davide's comment?2011-12-25
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    It's not clear from your question what framework you're working in. You speak of the delta "function", and you write $F(x)=\infty$. This seems to imply that you're considering $F$ as a function from the reals to the extended reals (extended by adding infinity). In that case, a) the Lebesgue integral over your function $F$ is $0$, not $1$ as it should be for anything that deserves to be called "the delta function", and b) $F$ is already fully defined by the second line in the question, so it's not clear what you mean when you say you define the delta function as $F$ with additional properties.2011-12-25
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    I get the impression that you may not be aware that what's often called the delta "function" is usually rigorously defined either as a distribution or as a measure. See [Wikipedia](http://en.wikipedia.org/wiki/Dirac_delta_function#Definitions) for these definitions. I don't see how to make sense of what you write in a framework dealing only with functions.2011-12-25
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    [This related answer](http://math.stackexchange.com/a/56684/6179) might help you approach @joriki's objections, which I fully share.2011-12-26
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    @joriki: Thanks for the reference. I realize that my question is not a very good one, sorry about that. I am new to the Dirac delta "function" and was a bit confused... I shall read the refs.2011-12-26

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