  In this paper we develop general formulas for the subdifferential of the
pointwise supremum of convex functions, which cover and unify both the compact
continuous and the non-compact non-continuous settings. From the non-continuous
to the continuous setting, we proceed by a compactification-based approach
which leads us to problems having compact index sets and upper
semi-continuously indexed mappings, giving rise to new characterizations of the
subdifferential of the supremum by means of upper semicontinuous regularized
functions and an enlarged compact index set. In the opposite sense, we rewrite
the subdifferential of these new regularized functions by using the original
data, also leading us to new results on the subdifferential of the supremum. We
give two applications in the last section, the first one concerning the
nonconvex Fenchel duality, and the second one establishing Fritz-John and KKT
conditions in convex semi-infinite programming.
