On almost \(\beta\)-continuous functions (Q1271976)

A subset \(A\) of a topological space \(X\) is called \(\beta\)-open if \(A\subset \text{Cl}(\text{Int}(\text{Cl}(A)))\). A function \(f:X\to Y\) is called almost \(\beta\)-continuous at \(x\in X\) if for every open neighbourhood \(V\) of \(f(x)\) there exists a \(\beta\)-open set \(U\) containing \(x\) such that \(f(U)\subset \text{Int} (\text{Cl}(V))\). Several characterizations, sufficient conditions, and necessary conditions of almost \(\beta\)-continuity are encountered.