Planar noninvertible maps have been studied recently by several authors such as Mira , Gardini , Cathala. Much of their work has been concentrated on analyzing some examples and making some conclusions on the properties of the maps . Our concern in this work is to study planar noninvertible continuously differentiable maps , we have proved two new theorems that are concerned with nonempty unbounded critical sets, and we have given two conjectures : one characterizes the attractor of the map when the critical set is parabola, and the other characterizes periodic points of the map when the critical set is a line or a hyperbola. We have studied some properties of such kind of maps in particular absorbing areas, invariant areas of such maps, and we have mentioned some of the known results of the subject and have given proofs of some of the results that have appeared in literature without a proof . Moreover we have studied some examples that show certain phenomena on absorbing areas . In our work , we have made use of the Matlab version 6.1 software to solve the discussed examples .