  We provide some necessary and sufficient conditions for a proper lower
semicontinuous convex function, defined on a real Banach space, to be locally
or globally Lipschitz continuous. Our criteria rely on the existence of a
bounded selection of the subdifferential mapping and the intersections of the
subdifferential mapping and the normal cone operator to the domain of the given
function. Moreover, we also point out that the Lipschitz continuity of the
given function on an open and bounded (not necessarily convex) set can be
characterized via the existence of a bounded selection of the subdifferential
mapping on the boundary of the given set and as a consequence it is equivalent
to the local Lipschitz continuity at every point on the boundary of that set.
Our results are applied to extend a Lipschitz and convex function to the whole
space and to study the Lipschitz continuity of its Moreau envelope functions.
