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

Article

## Abstract

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*∂*^{2}*f/∂x*^{2} are continuous on ℝ^{ n }*× 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.

## Key words

Envelope sup-type function second order directional derivative nondifferentiable function## Preview

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## Copyright information

© The Mathematical Programming Society, Inc. 1988