Softness, sleekness and regularity properties in nonsmooth analysis

We study continuity properties of tangent and normal cones to a closed subset of a Banach space. Such a study can be seen as a nonsmooth set theoretic analogue of the study of continuously differentiable functions. We give particular attention to the case where different concepts coincide, a desirable feature such properties are able to offer. We identify a wide class of sets and functions for which this property occurs.