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Let $n\in\mathbb N$, $a,b\in\mathbb R$, $a<b$, and a continuous function $f:(a,b)\to\mathbb R^n$ be given. Now the following holds true:

The set of accumulation points of $f$ at the boundary of the domain (say, at $a$) $$ M:=\{x\in \mathbb R^n\,:\,\exists_{(t_n)_{n\in\mathbb N}\subset(a,b)}\,\lim_{n\to\infty}t_n=a\text{ and }\lim_{n\to\infty}f(t_n)=x\} $$ is either empty, or contains one element, or is uncountable.

Spoken more intuitively, what this should mean is that if a continuous function cannot be continuously extended to the closure of its domain ("$|M|\neq 1$"), then it either diverges to infinity ("$M=\emptyset$") or is "of the type" $\sin(\frac1x)$ ("$|M|=\aleph_1$").

Does anyone know of a citable reference which features such a result? My guess is that if some Analysis book features this result, it'd just list it as an exercise, but this would be better than nothing, of course. Any answer or comment is appreciated!

For the sake of rigor, here's a possible proof:

Assume $|M|\geq 2$, that is, there exist $x_1,x_2\in M$ (with corresponding sequences $(s_n)_n$, $(t_n)_n$, respectively) as well as $R>0$ such that $x_2\not\in B_R(x_1)$. W.l.o.g. we may assume that $s_1>t_1>s_2>t_2>\ldots$ (e.g., by choosing appropriate subsequences). The idea is that given any ball which separates $x_1$ and $x_2$, $f$ has to cross the boundary infinitely many times so the intermediate value theorem plus compactness of balls/spheres in finite dimensions yields a new element of $M$.

Let $r\in(0,R)$. Now by definition of $M$ there exists $N\in\mathbb N$ such that $$ f(s_n)\in B_R(x_1) \quad\text{ and }\quad f(t_n)\not\in B_R(x_1) $$ for all $n\geq N$. This is where we use continuity of $f$ by applying the intermediate value theorem: for every $n\geq N$ there exists $\tau_n\in (t_n,s_n)$ such that the function intersects the sphere of radius $r$, that is, $f(\tau_n)\in S_r(x_1)$. This yields a sequence $(\tau_n)_{n\in\mathbb N}$ which converges to $a$ such that $(f(\tau_n))_{n\in\mathbb N}$ is a subsequence of $S_r(x_1)$. But we are in finite dimensions so $S_r(x_1)$ is compact -- thus there exists a subsequence $(\tau_{n_k})_{k\in\mathbb N}$ such that $(f(\tau_{n_k}))_{k\in\mathbb N}$ converges with limit also on $S_r(x_1)$, denoted by $x_r$. But this all we needed because $x_r\in M$ (by definition of $M$) and $r\in(0,R)$ was chosen arbitrarily, so by injectivity of $r\mapsto x_r$ the set $\{x_r:r\in(0,R)\}\subset M$ is uncountable. $\;\square$

Frederik vom Ende
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  • I don't have time to think about this now, but my answer to Name for multi-valued analogue of a limit, especially the references for cluster sets of complex-valued functions. Possibly relevant -- a glance at Collingwood's book (middle of p. 2) says that if $f$ is continuous on an open subset of the complex plane, then the cluster set of $f$ at a point where that subset is locally connected is either a point (hence has cardinality $1)$ or a continuum (hence has cardinality $\mathfrak c = 2^{\aleph_0}).$ (continued) – Dave L. Renfro Nov 05 '21 at 17:36
  • Incidentally, the cluster set of a function is always a closed set, and thus will have cardinality $\mathfrak c$ if it is uncountable. Thus, the cardinal number $\aleph_1,$ regardless of whether the continuum hypothesis is assumed, does not need to be mentioned. – Dave L. Renfro Nov 05 '21 at 17:37
  • Indeed, the result in Collingwood you mentioned seems to be just what I was looking for! (In his setting he allows for functions with complex or real domain). Interestingly, the result seems to hold for any locally connected metric space as co-domain -- and not just the case $\mathbb R^n$ which I mentioned above. (cont.) – Frederik vom Ende Nov 08 '21 at 11:58
  • The only restriction he seems to impose is that the range of $f$ has to be bounded, which is crucial for connectedness of the the cluster set (but not for it being a continuum). For unbounded functions the connectedness result should continue to hold if one works in the compactification of the co-domain (e.g., the cluster set could consist of two half-lines which are "connected at infinity"). Either way feel free to post the proposed reference as an answer and I will accept it! – Frederik vom Ende Nov 08 '21 at 12:01

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(edited and extended version of previously posted comments) Collingwood/Lohwater’s 1966 book The Theory of Cluster Sets says (middle of p. 2) that if $f:D \rightarrow {\mathbb C}$ and $D$ is an open subset of the complex plane ${\mathbb C},$ then the cluster set of $f$ at a point $z_0$ belonging to the (topological) boundary of $D$ such that $D$ is locally connected at $z_0$ is either a point (hence has cardinality $1)$ or a continuum (hence has cardinality $\mathfrak c = 2^{\aleph_0}).$ A similar observation $(D$ is a Jordan domain bounded by a simple closed curve) is made in the middle of p. 1 of Noshiro’s 1960 book Cluster Sets. See also p. 10 of James Reid Calhoun’s 1969 Ed.D. dissertation An Introduction to Cluster Set Theory, which unlike (as far as I know) the two books I previously mentioned, is freely available on the internet.

I don’t know much about these particular results, but I imagine they had been known many years before 1960, and I also suspect that there is a fair amount of research (some of which is likely covered in these references, but I haven’t checked them for this) into all kinds of refinements, generalizations, and variations of results related to the connectivity and continuum’ness of cluster sets.

Incidentally, the cluster set of a function is always a closed set, and thus will have cardinality $\mathfrak c$ if it is uncountable. Thus, the cardinal number ${\aleph_1},$ regardless of whether the continuum hypothesis is assumed, does not need to be mentioned.

Finally, my answer to Name for multi-valued analogue of a limit gives some pointers to the vast literature involving cluster sets. I have an annotated bibliography (chronologically ordered) of cluster set papers and books (currently 391 items, 280 pages) that I add to from time to time when my interest and free time are sufficient, if you’re interested (send me an email).

Dave L. Renfro
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