Local characterization of functions having the Cantor intermediate value property (Q1978958)

A function \(f:I\to \mathbb R\) is said to have the Cantor Intermediate Value Property (CIVP) if for each pair of points \(p,q\in I\) (\(p<q\), \(f(p)\neq f(q)\)) and for each Cantor set \(K\) between \(f(p)\) and \(f(q)\) there exists a Cantor set \(C\subset (p,q)\) such that \(f(C)\subset K\). If, in addition, \(f|_C\) is assumed to be continuous, \(f\) is said to have the strong CIVP. A function \(f\) is said to have the weak CIVP if for all \(p,q\in I\) (\(p<q\), \(f(p)\neq f(q)\)) there exists a Cantor set \(C\subset (p,q)\) such that \(f(C)\) is contained between \(f(p)\) and \(f(q)\). The notions of the CIVP and the strong CIVP at a point of an interval are introduced. Local characterizations of the CIVP and the strong CIVP are proved: A function \(f:I\to \mathbb R\) has the (strong) CIVP iff \(f\) has the (strong) CIVP at every point of the interval~\(I\). It is shown that no similar local characterization is possible for the weak CIVP.