ABSTRACT. For set-valued mappings in Euclidean space which are locally bounded, the notion of a Lipschitz mapping can be established using the Hausdorff distance between two sets. When considering mappings with unbounded images, the Hausdorff distance is not so appropriate. Instead, the idea of a sub-Lipschitz mapping is more natural. This talk will present a special kind of sub-Lipschitz behavior defined from the cosmic distance between sets. It will be shown that this type of sub-Lipschitz behavior is important when considering Hamilton-Jacobi equations, necessary optimality conditions for Bolza problems, and in viability theory.