Lipschitz-like Property of Subdifferential Mapping of Convex Continuous Functions on Hilbert Space

A Lipschitz-like property of subdifferential mappings of convex continuous functions on Hilbert spaces is investigated. It is shown that the subdifferential mapping of such a function admits a Lipschitz-like property at all points from a dense subset of its domain. In particular, Lipschitz-like properties of subdifferentials of distance functions from sets with convex complements are discovered.