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This post is quite long, since I wanted to include the necessary context. (Maybe I've put there too much.) Maybe you might prefer to look at the questions at the end of this post, first.

$\newcommand{\Filt}[1]{\mathcal{#1}}\newcommand{\Zobr}[3]{{#1}\colon{#2}\to{#3}}\newcommand{\FF}{\Filt F}\newcommand{\Invobr}[2]{{#1}^{-1}[#2]}$ If $\Zobr fXY$ is a map, $\Filt F$ is a filter on $X$ and $Y$ is a topological space, then we can define the notion of a limit (of a cluster point) of the function $f$ along the filter $\Filt F$.

A point $y\in Y$ is a limit of $f$ along $\FF$ if $$\Invobr fU=\{s\in S; f(s)\in U\} \in \FF$$ holds for every neighborhood $U$ of the point $y$.

A point $y\in Y$ is a cluster point of $f$ along $\FF$ if for every neighborhood $U$ of $y$ and for every $F\in\FF$ the intersection $\Invobr fU \cap F$ is non-empty.

Equivalently, the set of all $\FF$-cluster points can be described as $\bigcap_{F\in\FF} \overline{f[F]}$.

We get equivalent definitions if we talk about limit/cluster point of the filter base $f[\FF]=\{f[F]; F\in\FF\}$.

For reference, see e.g. Bourbaki: General Topology p.70, Dixmier: General Topology, Section 2.6 or Schaefer, Wolff: Topological Vector Spaces p.5. Some authors use the name adherence point instead of cluster point.

This type of limit was mentioned in this question.

In this question I am only interested in the special case of the definition of $\FF$-cluster point, when $X$ is a topological space and the filter $\FF$ is the neighborhood filter of some point $x\in X$. (I.e. the filter $\FF$ consists precisely of all neighborhoods of the point $x$.)

I.e. in this case, the set of all $\FF$-cluster points is $$\bigcap_{U\in\mathcal N_x} \overline{f[U]},$$ where $\mathcal N_x$ denotes the system of all neighborhoods of the point $x$ in $X$.

In this case the $\FF$-limit is precisely the limit of $f$ at the point $x$. An it seems very natural to call the $\FF$-cluster point of $f$ a cluster point of the function $f$ at the point $x$. In fact, such definition is used in Bourbaki, p.73.

For example, the cluster points of function $\sin\frac1x$ at $0$ are the points in the interval $[-1,1]$. If we apply the signum function to the preceding function, we get $\{-1,0,1\}$ as cluster points at $0$. (Here I am talking about cluster points of $f$ at a point, where the given function is not defined - but it's not very difficult to modify the above definition to this situation.)

Based on the above observations, this notion seems to be closely related to the continuity at a point. So I would expect that this notion and some results about it could appear in textbooks on real analysis at least for the special case of real functions or functions between metric spaces. (The generality in which I stated the definition above is, as far as I can say, common only in the texts directed more towards general topology.)

I've tried to Google a little and have a look into few books, which I thought might be good references on real analysis (Bruckner, Bruckner, Thomson: Real Analysis; Protter: Basic Elements of Real Analysis; Pugh: Real mathematical analysis). Often I stumbled upon the notion of cluster point of a sequence or cluster point of a set, but the only thing related to the above notion I was able to find was cluster value of a real function here. (The exercise I linked to says that for real functions all cluster values are between $\liminf_{a\to x} f(a)$ and $\limsup_{a\to x} f(a)$ an these two values are both cluster values. This means that the set of cluster values is a singleton if and only if both limit superior and limit inferior have the same value.)

As I did not find much results about this notion, I wonder whether I was searching for wrong name.

So my questions are:

  • Is the notion of cluster point of $f$ at $x$, which I tried to describe above, known under other names?
  • Do you know some references for basic results about this notion?
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I’ve not encountered it under any name. The notion of an $\mathscr{F}$-limit, yes; possibly the set of all cluster points of $f$ at $x$, though not with any special name; an individual cluster point of $f$ at $x$, no. –  Brian M. Scott Apr 15 '12 at 22:42

1 Answer 1

up vote 2 down vote accepted

This happens to be something I've accumulated a great deal of literature on over the past 20 years (several hundred papers). The literature is rather scattered and frequently in little known journals, and so various results are often rediscovered. Given what you've written and the time I have available right now, I'll limit myself to the following references. If you have more specific questions, I can probably point you to something in the literature, although it may take a day or two because all the papers and other materials I have are at home.

Charles Leonard Belna, Cluster sets of arbitrary real functions: A partial survey, Real Analysis Exchange 1 (1976), 7–20.

Ludek Zajicek, On cluster sets of arbitrary functions, Fundamenta Mathematica 83 (1973), 197-217.


Andrew M. Bruckner I and Brian S. Thomson, Real Variable Contributions of G. C. Young and W. H. Young, Expositiones Mathematicae 19 (2001), 337-358.


Brian S. Thomson, Real Functions, Lecture Notes in Mathematics #1170, Springer-Verlag, 1985, viii + 229 pages. [See the chapter on cluster sets.]

Jaroslav Lukes, Jan Maly, and Ludek Zajicek, Fine Topology Methods in Real Analysis and Potential Theory, Lecture Notes in Mathematics #1189, Springer-Verlag, 1986, x + 472 pages. [Various cluster set notions appear throughout.]

Cotinuous [sic] Functions and Limits [3-6 April 2007 sci.math thread.]



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Thanks a lot for your answer, which was very helpful. It seems that this notion, together with various similar notions was extensively studied. –  Martin Sleziak Apr 18 '12 at 17:20
@Martin Sleziak: I came across two papers by John Allan Eidswick you might be interested in. Both are in Proceedings of the Amer. Math. Soc. [39 (1973), pp. 163-168; 60 (1976), pp. 116-118] and freely available on the internet at ams.org/journals/proc/1973-039-01 and ams.org/journals/proc/1976-060-01 Together they show the undecidability in ZFC set theory of a rather natural problem involving various collections of cluster sets of a function $f:{\mathbb R}^2 \rightarrow {\mathbb R}$ at a fixed point for various collections of approach curves to that point. continued –  Dave L. Renfro Apr 19 '12 at 17:19
@Martin Sleziak: continued As far as I know, no one has done any follow-up work relating to what Eidswick has done, which makes it a topic that hasn't been "picked clean" by subsequent researchers, and thus there may be some "low-hanging fruit" for someone with the appropriate background (i.e. this is a specific example of something I was talking about in a comment at math.stackexchange.com/questions/131338 ). –  Dave L. Renfro Apr 19 '12 at 17:25

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