Quasicontinuous functions with everywhere discontinuous iterates
A set \(A\) of a topological space \((X,\tau)\) is said to be semi-open or quasi-open [\textit{N. Levine}, Am. Math. Mon. 70, 36--41 (1963; Zbl 0113.16304)] if \(A\subset\text{Cl}(\text{Int}(A))\). A function \(f: (X,\tau)\to (Y,\sigma)\) is said to be semicontinuous or quasicontinuous if for each \(x\in X\) and each open set \(V\) containing \(f(x)\), there exists an quasi-open set \(U\) containing \(x\) such that \(f(U)\subset V\). It is well-known that every quasicontinuous function has a dense -- indeed, residual -- set of points of continuity. In the present paper, the authors gives examples of two quasicontinuous functions whose second iterates are discontinuous everywhere.
scientific article; zbMATH DE number 5291262