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On the path Darboux property. (English) Zbl 1185.26003

Recall that a function \(f\) has the Darboux property if the image of an arbitrary connected set is connected. Bruckner and Ceder gave the definition of \(f\) to be Darboux at a point from right and left and Casazer showed that \(f\) has the Darboux property on an interval if and only if it is Darboux at each of its points. In this paper taking into consideration, a system of families of bilateral paths, the author introduces the notion of path Darboux property and proves its basic properties and investigates relationships between the path Darboux property and path continuity.

MSC:

26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
26A21 Classification of real functions; Baire classification of sets and functions
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