In this chapter we review some basic mathematical theory from non-smooth analysis [7,15, 44, 60, 61,128]. We begin with a brief introduction to sets and set-valued functions (Section 2.1). The notion of continuity of functions is relaxed in Section 2.2 to semi-continuity for set-valued functions and the notion of the classical derivative of smooth functions is extended in Section 2.3 to ... [Show full abstract] generalized differentials for non-smooth functions. Some definitions from convex analysis are reviewed in Section 2.4. Finally, an introduction is given to linear complementarity problems, which will prove to be useful when analyzing non-smooth dynamical systems.