Darboux functions with a perfect road (Q920213)

We say that a function f: \(<a,b>\to R\) has the property (i) B.; (ii) WCIVP (the weak Cantor intermediate value property) and (iii) P.R. (the property of perfect road) iff it holds (i) for any open interval (p,q) in \(<a,b>\) and any open interval E in R the set \((p,q)\cap f^{-1}(E)\) contains a perfect set whenever it is uncountable; (ii) for any \(p,q\in <a,b>\) for which \(p\neq q\) and \(f(p)\neq f(q)\) there exists a Cantor set C between p and q such that f(C) is between f(p) and f(q) and (iii) for any point \(x\in <a,b>\) there exists a perfect set P, called a perfect road for x, such that x is a bilateral limit point of P and f/P is continuous at x if \(x\in (a,b)\) and is an unilateral limit point of P and f/P is one-sided continuous at x if \(x\in \{a,b\}.\) The main result is the following theorem: If f: \(<a,b>\to R\) is a Darboux function, then the following statements are equivalent: (1) f has the property B., (2) f has the property WCIVP and (3) f has the property P.R.. There are given examples that the implications: \((1)\Rightarrow (2),\) \((2)\Rightarrow (3)\) and \((3)\Rightarrow (1)\) do not hold if f is not Darboux, and some remarks concerning also other properties of real functions.