On continuity and generalized continuity with respect to two topologies (Q1978974)

If \(\tau _1\) and \(\tau _2\) are two (not necessarily Hausdorff) topologies on \(X\) and \(\tau =\tau _1\cap \tau _2\), the relation of the continuity of a real-valued function \(f:X\to \mathbb R\) at an \(x\in X\) with respect to \(\tau \) to the continuity with respect to \(\tau _i\), \(i=1,2,\) is discussed. Similarly, the properties to be quasicontinuous at \(x\) and to be cliquish at \(x\) are also considered.