The upper and lower second order directional derivatives of a sup-type function

The purpose of this paper is to give a formula for expressing the second order directional derivatives of the sup-type function S(x) = sup{f(x, t); t ∈ T} in terms of the first and second derivatives of f(x, t), where T is a compact set in a metric space and we assume that f, ∂f/∂x and ∂²f/∂x² are continuous on ℝⁿ× T. We will give a geometrical meaning of the formula. We will moreover give a sufficient condition for S(x) to be directionally twice differentiable.