We generalize and improve the original characterization given by Valadier
[18, Theorem 1] of the subdifferential of the pointwise supremum of convex
functions, involving the subdifferentials of the data functions at nearby
points. We remove the continuity assumption made in that work and obtain a
general formula for such a subdiferential. In particular, when the supremum is
continuous at some point of its domain, but not necessarily at the reference
point, we get a simpler version which gives rise to the Valadier formula. Our
starting result is the characterization given in [11, Theorem 4], which uses
the epsilon-subdifferential at the reference point.Comment: 27 page