Points of continuity, quasicontinuity, cliquishness, and upper and lower quasicontinuity (Q2520021)

A real function \(f : X \to \mathbb R\) is said to be quasicontinuous (cliquish) at a point \(x \in \mathbb R\) if for each \(\varepsilon > 0\) and for each neighbourhood \(U\) of \(x\) there is a nonempty open set \(G \subset U\) such that \(|f(x) - f(y)| < \varepsilon\) for each \(y \in G\) (resp., \(|f(y) - f(z)| < \varepsilon\) for each \(y,z \in G\)). A function \(f : X \rightarrow \mathbb R\) is said to be upper (lower) quasicontinuous at \(x \in X\) if for each \(\varepsilon > 0\) and for each neighbourhood \(U\) of \(x\) there is a nonempty open set \(G \subset U\) such that \(f(y) < f(x) + \varepsilon\) (resp., \(f(y) > f(x) + \varepsilon)\) for each \(y \in G\). If \(C(f)\) is the set of all continuity points, \(Q(f)\) is the set of all quasicontinuity points, \(E(f)\) is the set of all points of both upper and lower quasicontinuity and \(E(f)\) is the set of all cliquishness points of \(f\), then it is well-known that \(C(f) \subset Q(f) \subset A(f)\), \(C(f)\) is \(G_\delta\), \(A(f)\) is closed, \(Q(f) \subset E(f)\) and \(A(F) \backslash C(f)\) is of first category. In paper [\textit{J. Ewert} and \textit{J. S. Lipiński} General topology and its relations to modern analysis and algebra VI, Proc. 6th Symp., Prague/Czech. 1986, Res. Expo. Math. 16, 177--185 (1988; Zbl 0642.54008)], the triplet \((C(f), Q(f), A(f))\) was characterised. Here the author tries to characterise the quadruplet \((C(f), Q(f), E(f), A(f))\). The author shows for example that \(E(f) \backslash A(f)\) is nowhere dense.