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Inifnite Limits

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Maury Barbato

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May 2, 2010, 11:32:34 AM5/2/10
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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

A N Niel

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May 2, 2010, 1:35:19 PM5/2/10
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In article
<482310758.58816.12728...@gallium.mathforum.org>,
Maury Barbato <maurizi...@aruba.it> wrote:

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.

Rob Johnson

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May 2, 2010, 2:52:15 PM5/2/10
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In article <482310758.58816.12728...@gallium.mathforum.org>,
Maury Barbato <maurizi...@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 <r...@trash.whim.org>
take out the trash before replying
to view any ASCII art, display article in a monospaced font

mister1729

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May 2, 2010, 5:17:19 PM5/2/10
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Inifnite Inifnite Inifnite Inifnite Inifnite Inifnite Inifnite Inifnite
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"Maury Barbato" <maurizi...@aruba.it> wrote in message
news:482310758.58816.12728...@gallium.mathforum.org...

Maury Barbato

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May 3, 2010, 4:29:11 AM5/3/10
to
A N Niel wrote:

> In article
> <482310758.58816.1272814384272.JavaMail.root@gallium.m

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

Thank you so much Niel, for your help.

Rob Johnson

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May 3, 2010, 11:42:04 AM5/3/10
to
In article <1118366644.61639.12728...@gallium.mathforum.org>,

Maury Barbato <maurizi...@aruba.it> wrote:
>A N Niel wrote:
>
>> In article
>> <482310758.58816.1272814384272.JavaMail.root@gallium.m
>> athforum.org>,
>> Maury Barbato <maurizi...@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?!

Indeed. Although in my (erroneous) counterexample, I forgot to
define f(q_n) = n, and the function is finite almost everywhere,
the remaining set of measure 0 on which the sum is unbounded must
not be empty. This indicates that there may be some interesting
number theoretic results involving approximations here somewhere.

Dave L. Renfro

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May 4, 2010, 10:05:43 AM5/4/10
to
Maury Barbato 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?
> Thank you very much for your attention.

Others have responded (I didn't see this post until a few
minutes ago), but I thought it would be useful to mention
a possibly subtle distintion. A function can be unbounded in
every open interval (in fact, a Baire 2 function can have a
graph that is dense in the plane), so we can have

lim-sup_{x --> y} f(x) = +infinity

for every real number y, even for some fairly nice functions
(e.g. certain Baire 2 functions). For some examples, see
the following posts:

sci.math: Exotic functions (elementary)
5 & 6 October 2006
http://groups.google.com/group/sci.math/msg/98d0bb02228bd4bd
http://groups.google.com/group/sci.math/msg/4016347301a71140

On the other hand, the property you require is that the limit
exists (as +infinity) at the points y, and this can only be
the case for very small sets. For one thing, the set has to be
countable. For another thing, the set cannot be dense in any
open interval. In fact, the set cannot even be dense in any
nonempty perfect set. For example, the set cannot be a dense
subset of the thin-and-nowhere-dense Cantor middle thirds set.
The sets E that have the property you're looking at (namely,
there exists a function f:R --> R such that f has an infinite
limit at y if and only if y belongs to E) happen to be the
scattered subsets of R. These have several characterizations,
one of which is that they are the collection of countable G_delta
subsets of R. For other characterizations and references about
scattered sets, see the posts in the sci.math thread "Scattered
sets are G-delta" (of which an excerpt from one post in that
thread is copied below).

--------------------------------------------------------
--------------------------------------------------------

sci.math: Scattered sets are G-delta
26 May 2008
http://groups.google.com/group/sci.math/msg/6098777da158cb95

Finally, another poster mentioned "A subset S of R is
scattered iff there is some map f:R -> R such that
S = {x| lim as y->x f(y) is + infinity}.", a result
I've previously posted about. In case someone wants
references for this (and it's likely the result
appears in papers by one or more of the authors I
mentioned above where Frechet is named) --->

Burnett Meyer, "On restricted functions", American
Mathematical Monthly 62 #1 (January 1955), 29-30.

Robert Judson Bumcrot and Mark Sheingorn, "Variations
on continuity: Sets of infinite limit", Mathematics
Magazine 47 #1 (Jan./Feb. 1974), 41-43.

Janos T. Toth and Laszlo Zsilinszky, "On the class
of functions having infinite limit on a given set",
Colloquium Mathematicum 67 (1994), 177-180.

Tomasz Natkaniec, "On sets determined by limits
of a real function", Scientific Bulletin of Lodz
Technical University [= Zeszyty Naukowe Politechniki
Lodzkiej], Matematyka 27 #719 (1995), 95-104.

--------------------------------------------------------
--------------------------------------------------------

sci.math: Algebraic numbers are not G_delta ?
30 November 2007
http://groups.google.com/group/sci.math/msg/0fb6110e86c6e017

Indeed, any countable G_delta set is "maximally small"
in a certain topological sense. More specifically,
a countable G_delta set in R (or in any complete
separable metric space) is nowhere dense relative
to every nonempty perfect subset of R. This means,
for instance, that not only the set itself but
also the closure of the set fails to be dense
in any portion of any perfect set, regardless
of how small/thin the perfect set is. Moreover,
this characterizes the countable G_delta sets
(often called "scattered sets" in the literature),
since any set that is nowhere dense relative to
every nonempty perfect set is a countable G_delta
set. An example of a situation in which scattered
sets arise is the set of all points where a function
has an infinite limit. Given f:R --> R, the set
{y: lim(x --> y) f(x) is +infinity} is scattered.

In fact, any scattered subset of R can be realized
in this way for some function f. The same is true
if the limit operation is restricted to being a
left limit or to being a right limit, and/or
+infinity being replaced with -infinity or with
either infinity. However, if "infinity" is replaced
with a specified finite value, the set may no
longer be countable (e.g. a constant function),
although it will still be G_delta.

Being scattered is a type of smallness that fits
between isolated sets (all of which are scattered,
but a convergent sequence union its limit point is
a scattered set that isn't an isolated set) and
the nowhere dense sets (every scattered set is
nowhere dense, but a nonempty nowhere dense perfect
set -- the Cantor set, for instance -- is a nowhere
dense set that isn't scattered.

--------------------------------------------------------
--------------------------------------------------------

Dave L. Renfro

Maury Barbato

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May 4, 2010, 11:16:53 AM5/4/10
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Dave L. renfro wrote:

Thank you very very much, Dave, for the usual generous
plenty of references. Your knowledge of mathematics is
encyclopaedic ...

Friendly Regards,
Maury Barbato

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