Some examples on continuous restrictions
In the present paper the author raised the following question: Are there classes of functions, f: \(R\to R\) (R - the real line), not having the property of Baire, for which there exists a dense subset A of R with A uncountable such that the restriction \(f| A\) is continuous? Ingenious examples show that the Lebesgue measurable functions and the connected functions (i.e., the graph is a connected set), fail to have this property. A function f: \(R\to R\) is said to be countably decomposable into continuous functions if there exist countably many disjoint sets \(A_ n\), \(n=1,2,...\), whose union is R such that \(f| A_ n\) is continuous for each n. It is shown that there exists a bounded, approximately continuous, lower-semi-continuous function which is not countably decomposable into continuous functions. This result improves a result of \textit{R. O. Davies} [Čas. Pěstovani Mat. 98, 398-399 (1973; Zbl 0272.26003)].
countable decomposition into continuous functions
scientific article; zbMATH DE number 3845876