Functional connectedness and Darboux property of multivalued functions. (English) Zbl 0847.54018

Czarnowska, J.

Period. Math. Hung. 26, No. 2, 101-110 (1993).

In [A. M. Bruckner and J. G. Ceder, Jahresber. Dtsch. Math.-Ver. 67, 93-117 (1965; Zbl 0144.30003)] and [J. M. Jastrzȩbski and J. M. Jȩdrzejewski, Zesz. Nauk Politech. Śl., Mat.-Fiz. 853(48), 73-80 (1986; Zbl 0777.26008)] the notions of Darboux points, functional connectedness points and global functional connectedness of a real function of a real variable were introduced. In those papers it was also proved that a function \(f:\mathbb{R}\to\mathbb{R}\) is a Darboux (functionally connected) function if and only if it is Darboux (functionally connected) at every point of its domain. In [H. Rosen, Fundam. Math. 89, 265-269 (1975; Zbl 0316.26004)] it is shown that the set of Darboux points of a function \(f : \mathbb{R} \to \mathbb{R}\) is a \(G_\delta\)-set. The same result for functional connectedness points was proven in [J. M. Jastrzȩbski and J. M. Jȩdrzejewski, loc. cit.]. The results included in this paper generalize those mentioned above onto the case of multivalued functions.

MSC:

54C60

Set-valued maps in general topology

Keywords:

Darboux property; functional connectedness points

Citations:

Zbl 0144.30003; Zbl 0777.26008; Zbl 0316.26004

References:

[1]	A. M. Bruckner andJ. G. Ceder, Darboux continuity,Jahresbericht d. Deutschen Mathem.-Vereinigung 67 (1965), 93–117. · Zbl 0144.30003
[2]	J. Czarnowska andG. Kwiecińska, On the Darboux property of multivalued functions,Demonstr. Math. XXV (1992), No. 1–2, 193–199. · Zbl 0765.54010
[3]	J. M. Jastrzebski andJ. M. Jedrzejewski, Functionally connected functions,Zeszyty Naukowe Politechniki Slaskiej, Mat.-Fiz. 48 (1986), 73–79.
[4]	H. Rosen, Connectivity points and Darboux points of real functions,Fund. Math. 89 (1975), 265–269. · Zbl 0316.26004