Functions in Digital and Discrete Space

Abstract

In this chapter, we introduce the concepts of digital functions and their basic properties. We start with digital “continuous” functions by Rosenfeld, then on to gradually varied functions proposed by Chen. We also present the necessary and sufficient condition of existence for gradually varied interpolations. For practical uses of these concepts, algorithm design is also a central topic in this chapter. We have designed different algorithms for gradually varied extension. Digital continuous functions and gradually varied functions are similar, but have some differences. Digital continuous functions are mainly defined on digital spaces and gradually varied functions are defined on any discrete space. At the end of the chapter, we discuss the mathematical foundations for gradually varied functions through constructive analysis.