A note on Saleh ’ s paper ‘ Almost continuity implies closure continuity ’ ∗

The first two implications are trivial. In [2], the second author showed that every almost α-continuous function is η-continuous. In [1], Dickman, Porter and Rubin proved that every η-continuous is θ-continuous and hence weakly continuous. Although θ-continuous functions behave, in general, nicely, they may cause some unexpected problems. For example, if f :X → Y is θ-continuous, then f :X → f(X) is not necessarily θ-continuous. Also, the set of all points of continuity of f :X → Y may be dense in X and f may not be weakly continuous in either of those points. We show that with the following example: Example. Consider the classical Dirichlet function f : (R, τd) → (R, τd), where R is the real line with the density topology τd: