In this question I asked if there was an analogue of connectedness which applied to dense orders which were not required to be complete. Between them, Noah and Joel showed that every (infinite) topological space is the continuous image of some dense order. But a couple of new questions arose:

As Noah suggests, what happens if we restrain the continuous map f to be injective? It seems to me it's sufficient to restrict f not to be locally constant at any point.

Is there some sort of local property P, stronger than continuity, which applies to the map, in the sense that every x in the dense order X must have a neighbourhood U such that f(U) has property P? For example, in Noah and Joel's construction the map is locally constant, so that a local form of connectedness still holds.