This paper is concerned with first-order optimality conditions for nonsmooth extremal problems. The author first studies positively homogeneous functions (differences of sublinear functions) which are used later in local approximations and difference convex domains. A theorem stating a necessary and sufficient condition for a positively homogeneous function \(\phi\) to belong to the linear space of difference sublinear functions on \(E_ n\) is proved. Furthermore it is proved that every closed cone C is associated with a difference sublinear function \(\phi\) such that \(C=\{x| \phi (x)\leq 0\}\). After some further remarks on quasidifferentiable functions and approximations using difference sublinear functions the last section concerns first-order optimality conditions and the theory of quasidifferential calculus developed earlier.