On the points of bilateral quasicontinuity of functions (Q1332591)

Let \(C(f)\), \(BQ(f)\), \(Q^ -(f)\), \(Q^ +(f)\), \(Q(f)\), and \(A(f)\) denote the set of all continuity, bilateral quasicontinuity, left-hand sided quasicontinuity, right-hand sided quasicontinuity, quasicontinuity, and cliquishness points of the function \(f: \mathbb{R}\to\mathbb{R}\), respectively. The triplet \((C(f), BQ(f), A(f))\) has been characterized by J. Ewert and J. Lipiński. The sixtuple \[ (C(f), BQ(f), Q^ -(f), Q^ +(f), Q(f), A(f)) \] is characterized in the present paper. For a given sixtuple \((C,D,D_ 1,D_ 2,E,A)\) the following conditions are equivalent: 1. \((C,D,D_ 1,D_ 2,E,A)\) is equal to \((C(f), BQ(f), Q^ -(f), Q^ +(f), Q(f), A(f))\) for some \(f: \mathbb{R}\to\mathbb{R}\); 2. \(C\subset D= D_ 1\cap D_ 2\subset D_ 1\cup D_ 2= E\subset A\), \(C\) is a \(G_ \delta\) set, \(A\) is closed, \(A\backslash C\) is of the first category and \(E\backslash D\) is countable.