Concerning extendable connectivity functions (Q1084196)

This paper is a survey concerning extendable connectivity functions. \textit{J. Stallings} [Fundam. Math. 47, 249-263 (1959; Zbl 0114.391)] asked the question: ''If \(I=[0,1]\) is embedded in \(I^ 2\) as \(I\times 0,\) can a connectivity function \(I\to X\) be extended to a connectivity function \(I^ 2\to X?\) Negative answers to this question were given by \textit{J. L. Cornette} [Fundam. Math. 58, 183-192 (1966; Zbl 0139.406)] and \textit{J. H. Roberts} [Fundam. Math. 57, 173-179 (1965; Zbl 0139.406)]. Since almost continuous functions \(I\to I\) are connectivity functions, a natural question arises. Question 0. Can an almost continuous function \(I\to I\) be extended to a connectivity function \(I^ 2\to I?\) An example [the author and \textit{F. Roush}, Topology, Proc. Conf., Vol. 7, No.1, Annapolis/Md. 1982, 55-62 (1982; Zbl 0518.26001)] of an almost continuous function \(I\to I\) was constructed and shown to be not extendable [the author and \textit{F. Roush}, Topology, Proc. 10, No.1, 75-82 (1985; Zbl 0595.54001)]. An example [the author and \textit{F. Roush}, Real Anal. Exch., to appear] of a connectivity function \(g:I^ 2\to I\) that has the property that for some \(p\in I,\) \(g| (I\times p)\) does not have property (s) was announced. Some of the other questions are as follows. Question 2. Does there exist a Baire class 1 connectivity function \(I\to I\) that can not be extended to a connectivity function \(I^ 2\to I?\) Answered in the negative by \textit{J. Brown, P. Humke}, and \textit{M. Laczkovich} [Proc. Am. Math. Soc., to appear]. Question 3. If \(g:I^ 2\to I\) is a connectivity function and \(f:I\to I\) is a function such that \(f\circ g:I^ 2\to I\) is a connectivity function, is f continuous except perhaps at 0 or 1? Proved by H. Rosen for g a Darboux function and will be contained in a paper with R. G. Gibson. Question 5. Is it true that if \(f:I\to I\) can be extended to a connectivity function \(I^ 2\to I,\) then f can be extended to a connectivity function \(g:I^ 2\to I\) sucht that g is continuous on the complement of \(I\times 0?\) This is true and will be contained in a paper by R. G. Gibson and F. Roush.