Inifnite Limits

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From: Maury Barbato on 2 May 2010 07:32

---

Hello,
there is some real function f:[0,1]-> R, such that
for some dense subset E of [0,1], we have

lim_{x -> y} f(x) = + infinity,

for every y in E?

I think the answer is yes, but I don't see how
to construct such a function. Some idea?
Thank you very much for your attention.

My Best Regards,
Maury Barbato

From: A N Niel on 2 May 2010 13:35

---

In article
<482310758.58816.1272814384272.JavaMail.root(a)gallium.mathforum.org>,
Maury Barbato <mauriziobarbato(a)aruba.it> wrote:

> Hello,
> there is some real function f:[0,1]-> R, such that
> for some dense subset E of [0,1], we have
>
> lim_{x -> y} f(x) = + infinity,
>
> for every y in E?
>
> I think the answer is yes, but I don't see how
> to construct such a function. Some idea?
> Thank you very much for your attention.
>
> My Best Regards,
> Maury Barbato

Let's see. For each n, f(x) > n at least on a dense open set E_n. But
the intersection of all these E_n is empty. No, not possible.

From: Rob Johnson on 2 May 2010 14:52

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In article <482310758.58816.1272814384272.JavaMail.root(a)gallium.mathforum.org>,
Maury Barbato <mauriziobarbato(a)aruba.it> wrote:
>there is some real function f:[0,1]-> R, such that
>for some dense subset E of [0,1], we have
>
>lim_{x -> y} f(x) = + infinity,
>
>for every y in E?
>
>I think the answer is yes, but I don't see how
>to construct such a function. Some idea?

Let { q_k } be an enumeration of the rationals and define

g(x) = (x^2 + x^4)^{-1/3}

Note that g is positive and

|\oo
G = | g(x) dx < oo
\|-oo

In fact, G = Gamma(1/6)^2/Gamma(1/3). Furthermore,

lim g(x) = oo
x->0

Now, define

oo
--- -k
f(x) = > 2 g(x - q_k)
---
k=1

We must have that for each rational q

lim f(x) = oo
x->q

However,

|\oo
| f(x) dx = G < oo
\|-oo

so f is finite almost everywhere.

Rob Johnson <rob(a)trash.whim.org>
take out the trash before replying
to view any ASCII art, display article in a monospaced font

From: mister1729 on 2 May 2010 17:17

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"Maury Barbato" <mauriziobarbato(a)aruba.it> wrote in message
news:482310758.58816.1272814384272.JavaMail.root(a)gallium.mathforum.org...
> Hello,
> there is some real function f:[0,1]-> R, such that
> for some dense subset E of [0,1], we have
>
> lim_{x -> y} f(x) = + infinity,
>
> for every y in E?
>
> I think the answer is yes, but I don't see how
> to construct such a function. Some idea?
> Thank you very much for your attention.
>
> My Best Regards,
> Maury Barbato

From: Maury Barbato on 3 May 2010 00:29

---

A N Niel wrote:

> In article
> <482310758.58816.1272814384272.JavaMail.root(a)gallium.m
> athforum.org>,
> Maury Barbato <mauriziobarbato(a)aruba.it> wrote:
>
> > Hello,
> > there is some real function f:[0,1]-> R, such that
> > for some dense subset E of [0,1], we have
> >
> > lim_{x -> y} f(x) = + infinity,
> >
> > for every y in E?
> >
> > I think the answer is yes, but I don't see how
> > to construct such a function. Some idea?
> > Thank you very much for your attention.
> >
> > My Best Regards,
> > Maury Barbato
>
> Let's see. For each n, f(x) > n at least on a dense
> open set E_n. But
> the intersection of all these E_n is empty. No, not
> possible.

Yes, using Baire's Theorem ... how didn't I think
about it before?!

Thank you so much Niel, for your help.
My Best Regards,
Maury Barbato

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