A continuous tale on continuous and separately continuous functions
The main objective of this survey research paper is to discuss the history and actual state of continuous and separately continuous functions problem from a real analysis perspective. The exposition includes as well some original results; the basic ones are Theorem 1. For every \(n\geq 2\), a function \(f:\mathbb{R}^n\to \mathbb{R}\) is continuous, provided it is either \((C_n^1)^*\)-continuous or \((C_n^1)\)-continuous. Theorem 2. For every \(n\geq 2\), there exists a Baire class 1 function \(h:\mathbb{R}\to \mathbb{R}^{n-1}\) such that \(T(h)\)-continuity of \(f:\mathbb{R}^n\to \mathbb{R}\) implies its continuity. Finally, simplified versions are given for old results in the area, and several open problems are formulated.
linearly and hyperplane continuous functions
scientific article; zbMATH DE number 6848915