Continuity is often taken as a defining assumption of topological dynamical systems. In the last years, however, increasing attention has been paid to the investigation of highly discontinuous maps from the point of view of topological dynamics. Continuity has been relaxed in ways allowing a dense set of discontinuous points: for instance Darboux, Baire 1 and 2, almost-continuous and quasi-continuous maps have been considered. There are good reasons, beyond the mere search for generality, for this research direction. The talk will discuss these reasons and will describe a particular class of densely discontinuous maps, i.e. those generated by the symbolic action of erasing substitutions on the binary expansion of reals in the unit interval.