The uniform limit of connectivity functions
In this paper a Darboux function \(I\to I\) which is not the uniform limit of a sequence of connectivity functions is constructed, where \(I=[0,1].\) Also the following proposition is proved. Proposition A. Let X be a metric space. Then the uniform limit of a sequence of peripherally continuous functions \(X\to Reals\) is peripherally continuous. As corollaries to Proposition A the following propositions are given. Proposition B. Let \(f_ m\) be a sequence of real-valued functions defined on an interval. If each \(f_ m\) is a Baire class 1 Darboux function and \(f_ m\) converges to f uniformly, then f is a Baire class 1 Darboux function. Proposition C. Let \(f_ m:I^ n\to I\) be a sequence of functions, where \(n\geq 2.\) If each \(f_ m\) is a connectivity function and \(f_ m\) converges to f uniformly, then f is a connectivity function. The following question is stated. Does there exist a connectivity function \(f:I\to I\) that is not the uniform limit of a sequence of almost continuous functions \(f_ m:I\to I\) ?
scientific article; zbMATH DE number 3977308