This is a continuation of the author’s paper [Topol. Proc. 28, 229–239 (2004; Zbl 1081.54030)]. In the paper under review, the author adds a statement (part (1) in the following theorem) to the main theorem in the cited paper:
Theorem 7. For a continuum \(X\), the following three statements are equivalent:
(1) every connected real-valued function on \(X\) is a local connectivity function;
(2) every connected real-valued function on \(X\) is a connectivity function;
(3) every connected real-valued function on \(X\) is a Darboux function;
(4) \(X\) is a dendrite such that each arc in \(X\) contains only finitely many branch points of \(X\).
The author introduces a nice concept called \(\varepsilon\)-splitting set, and uses it to prove the following characterizations:
Theorem 13. Let \(X\) be an arc-like continuum. Then the following statements are equivalent:
(1) every real-valued local connectivity function on \(X\) is a connected function;
(2) every real-valued local connectivity function on \(X\) is a Darboux function;
(3) every real-valued local connectivity function on \(X\) is a connectivity function;
(4) \(X\) is an arc.
Theorem 17. Let \(X\) be a circle-like continuum. Then the following statements are equivalent;
(1) every real-valued local connectivity function on \(X\) is a Darboux function;
(2) every real-valued local connectivity function on \(X\) is a connectivity function;
(3) \(X\) is a simple closed curve.
A key result of this paper, which is used in most of the results is:
Lemma 1. Let \(X\) be a Peano continuum. Then every local connectivity function from \(X\) to a topological space \(Y\) is a connected function.
With these results the author gives partial answers to questions of J. Stallings, posed in 1959.