×

On some properties of sets blocking almost continuous functions. (English) Zbl 1034.26003

A closed set \(B \subset I\times {\mathcal R}\) is blocking if \(f \cap B = \emptyset \) for at least one function \(f:I \to {\mathcal R}\) and \(g \cap B \neq \emptyset \) for each continuous function \(g:I \to {\mathcal R}\). For a blocking set \(B \subset I\times {\mathcal R}\) let \({\mathcal E} = \{ [a,b];\text{there\;\;is\;\;a\;\;continuous}\;\; h:[0,a] \to {\mathcal R}\;\;\text{with}\;\;h(a)=b\;\;\text{and}\;\;h\cap B = \emptyset \} \) and let \({\mathcal N}(B) = (I\times {\mathcal R}) \setminus (B\cup {\mathcal E}(B))\). Using these operators the author gives new proofs of some classical theorems and proves some new theorems on the almost continuity and on the uniform limits of sequences of almost continuous functions.

MSC:

26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
26A21 Classification of real functions; Baire classification of sets and functions
54C30 Real-valued functions in general topology
PDFBibTeX XMLCite