Functions

Abstract

This chapter deals with Function Theory from a general perspective. Set Theory, relations and operations are overviewed with the aim of gathering all the necessary tools to develop limits and continuity, as well as Differential Calculus and Integral Calculus. An introduction to Set Theory, predicative logic and axiomatic systems, with particular emphasis to the Zermelo-Fraenkel axiomatic system, is provided in the first section of this chapter to make this book as selfcontained as possible. The second section deals with relations between families of sets, with a special emphasis on binary internal relations, such as equivalence relations and order relations. Lattices, filters, ideals, bornologies, and nets will also be overviewed, as they constitute essential topological tools, which are fundamental to develop limits and continuity. The third and final section of this first chapter on functions deals with multiary operations. Universal algebras (sets endowed with multiary operations) are described and studied. A strong focus will also be put on binary relations and sets endowed with them, such as groups, rings, and modules. Other, less common, universal algebras like effect algebras and Boolean algebras are also considered. Topologies will be defined as universal subalgberas. Metric spaces and normed groups are also defined as examples of universal algebras.