For the family of double standard maps we investigate the structure of the space of parameters a when and when. In the first case the maps have a critical point, but for a set of parameters of positive Lebesgue measure there is an invariant absolutely continuous measure for. In the second case there is an open non-empty set of parameters for which the map is expanding. We show that as, the set accumulates on many points of in a regular way from the measure point of view.