Suppose that f and h are convex functionals on a convex set S in Euclidean space with

Question:

Suppose that f and h are convex functionals on a convex set S in Euclidean space with f differentiable at x0 and
f(x0) = h(x0) and f(x) ≥ h(x) for every x ∈ S (31)
Then h is differentiable at x0, with Dh[x0] = Df[x0].
When X ⊆ ℜn, the derivative can be represented by the gradient, which provides the following useful alternative characterization of a convex function.