UPPER AND LOWER SECOND ORDER DIRECTIONAL DERIVATIVES OF A SUP-TYPE FUNCTION.

The purpose of this paper is to give a formula for expessing the second order directional derivatives of the sup-type function S(x) equals sup left brace f(x,t); epsilon TRTBC 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, PARTIAL T right brace in terms of the first and second drivatives of f(x,t), where T is a compact set in a metric space and we assume that f, PARTIAL f/ PARTIAL x and PARTIAL **2**4f/ PARTIAL x**2 are continuous on R double prime multiplied by 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.