Introduction to the Time Scales Calculus

In this chapter we introduce some basic concepts concerning the calculus on time scales that one needs to know to read this book. Most of these results will be stated without proof. Proofs can be found in the book by Bohner and Peterson [86]. A time scale is an arbitrary nonempty closed subset of the real numbers. Thus $$ \mathbb{R},\mathbb{Z},\mathbb{N},\mathbb{N}_0 , $$ i.e., the real numbers, the integers, the natural numbers, and the nonnegative integers are examples of time scales, as are $$ [0,1] \cup [2,3],[0,1] \cup \mathbb{N} $$ , and the Cantor set, while $$ \mathbb{Q},\mathbb{R}\backslash \mathbb{Q},\mathbb{C}(0,1), $$ i.e., the rational numbers, the irrational numbers, the complex numbers, and the open interval between 0 and 1, are not time scales. Throughout this book we will denote a time scale by the symbol $$ \mathbb{T} $$ . We assume throughout that a time scale $$ \mathbb{T} $$ has the topology that it inherits from the real numbers with the standard topology.