A generalised topology is a collection of subsets of a given nonempty set containing the empty set and arbitrary unions of the elements in the collection. By using the concept of hereditary classes, a generalised topology can be extended to a new one, called a generalised topology via a hereditary class. We study continuity on generalised topological spaces via hereditary classes in various situations.

We study the existence of continuity points for mappings $f\,:\,X\,\times \,Y\,\to \,Z$ whose $x$-sections $Y\,\backepsilon \,y\,\to \,f\left( x,y \right)\,\in \,Z$ are fragmentable and $y$-sections $X\,\backepsilon \,x\,\to \,f\left( x,y \right)\,\in \,Z$ are quasicontinuous, where $X$ is a Baire space and $Z$ is a metric space. For the factor $Y$, we consider two infinite “point-picking” games ${{G}_{1}}\,\left( y \right)$ and ${{G}_{2}}\,\left( y \right)$ defined respectively for each $y\,\in \,Y$ as follows: in the $n$-th inning, Player I gives a dense set ${{D}_{n}}\,\subset \,Y$, respectively, a dense open set ${{D}_{n}}\,\subset \,Y$. Then Player II picks a point ${{y}_{n}}\,\in \,{{D}_{n}}$; II wins if $y$ is in the closure of $\left\{ {{y}_{n}}\,:\,n\,\in \,\mathbb{N} \right\}$, otherwise I wins. It is shown that (i) $f$ is cliquish if II has a winning strategy in ${{G}_{1}}\,\left( y \right)$ for every $y\,\in \,Y$, and (ii) $f$ is quasicontinuous if the $x$-sections of $f$ are continuous and the set of $y\,\in \,Y$ such that II has a winning strategy in ${{G}_{2}}\,\left( y \right)$ is dense in $Y$. Item (i) extends substantially a result of Debs and item (ii) indicates that the problem of Talagrand on separately continuous maps has a positive answer for a wide class of “small” compact spaces.

In an $\text{H}$-closed, Urysohn space, disjoint $\text{H}$-sets can be separated by disjoint open sets. This is not true for an arbitrary H-closed space even if one of the $\text{H}$-sets is a point. In this paper, we provide a systematic study of those spaces in which disjoint $\text{H}$-sets can be separated by disjoint open sets.

We provide vector analogues of the classical interpolation theorems for lower semicontinuous functions due to Dowker and to Hahn and Katetov-Tong.

The paper discusses some consequences of weak monotonicity for connected maps in relation to essential connectedness of a space. The first main result gives conditions under which the image by a connected map of an essentially connected space is essentially connected. The second is that, for a connected mapping of a connected, 1 .c. space to a WLOTS-wise and essentially connected space, w-monotonicity implies monotonicity. The remainder of the paper discusses continuity properties of connected, w-monotone mappings with WLOTS-wise and essentially connected range.

A function f:X → Y is almost continuous if for every x ∊ X and for each open set V ⊂ Y containing f(x), Cl(f-l(V)) is a neighborhood of x. Various conditions are given that guarantee that an almost continuous function is continuous. The main theorem states that if f:X → Y is almost continuous with a closed graph (closed in X × Y) and X and Y are complete metric spaces, then f is continuous.