Inifnite Limits
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From: Rob Johnson on 3 May 2010 11:42 In article <1118366644.61639.1272875381739.JavaMail.root(a)gallium.mathforum.org>, Maury Barbato <mauriziobarbato(a)aruba.it> wrote: >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?! 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. 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: Dave L. Renfro on 4 May 2010 10:05 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
From: Maury Barbato on 4 May 2010 07:16 Dave L. renfro wrote: > 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/98d0bb0222 > 8bd4bd > http://groups.google.com/group/sci.math/msg/4016347301 > a71140 > > 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/6098777da1 > 58cb95 > > 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/0fb6110e86 > c6e017 > > 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 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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