In this paper, we investigate the dynamical behaviors of a novel class of Cohen-Grossberg neural networks with not only mixed time delays, i.e., time varying delays and distributed delays, but also discontinuous activations, which may be unbounded or nonmonotonic. Some sufficient conditions are derived to ensure the existence, uniqueness of the equilibrium, global exponential asymptotic stability of the solution and the associated output of the solution converging to the output equilibrium point in measure by using the Leray-Schauder alternativ...