Darboux like functions
The article contains a survey of research that has been done concerning the intermediate value property of real functions of a real variable and related properties. The following classes of functions are discussed: Darboux functions, peripherally continuous functions, connectivity functions, almost continuous functions in the sense of Stallings, extendable functions, functions with a perfect road, functions having the (weak, strong) Cantor intermediate value property, functions with property~\(B\). Functions from these classes are called briefly Darboux like functions. The survey includes Darboux like functions in the first class of Baire, Borel, Lebesgue and Marczewski measurability of Darboux like functions, Darboux like properties of Sierpiński-Zygmund functions, additive functions and quasi-continuous functions, characterizations of Darboux like functions, Darboux like functions of several variables, algebraic operations. Some open questions are posed.
scientific article; zbMATH DE number 1450282