On two generalizations of the Darboux property
In this paper the author investigates two classes of functions: \(U\) -- the family of all functions \(f:I\to\mathbb{R}\) such that \([\inf\{f(x):x\in J\},\sup\{f(x):x\in J\}]\subset\overline{f(J)}\) for each interval \(J\subset I\); \(U_ 0\) -- the family of all functions \(f:I\to\mathbb{R}\) such that \([\inf\{f(x):x\in J\},\sup\{f(x):x\in J\}]\subset\overline{f(J\backslash A)}\) for each interval \(J\subset I\) and each set \(A\) of cardinality less than \(c\). The author proves that: -- Any function \(f:\mathbb{R}\to\mathbb{R}\) is the product of two functions from \(U\). -- For a nontrivial connected subset \(A\) of \(\mathbb{R}\), if \(f:A\to\mathbb{R}\), then \(f\in U_ 0\) iff \(f\) is a preconnectivity function. -- The class \(U_ 0\;(U)\) is closed with respect to quasiuniform Arzelà-Gagaeff-Alexandrov convergence. -- \(A\subset(0,1)\) is a stationary set for \(U\;(U_ 0)\) iff its complement is at most countable (does not contain a set which is bilaterally dense in itself). -- The function \(f:\mathbb{R}\to\mathbb{R}\) is \(\nu\)-continuous iff \(f\in U\).
quasiuniform Arzelà-Gagaeff- Alexandrov convergence
scientific article; zbMATH DE number 68372