ABSTRACT
Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.
Bertrand Russell
Much of the material of this chapter should be familiar to students who have completed a “transition course” to higher mathematics, and for this reason many of the proofs are omitted. We define the general concept of a function and we assume the reader is well acquainted with real-valued functions of a single real variable, and even somewhat familiar with real-valued and vector-valued functions of a vector. We introduce the important class of complex functions in a bit more detail, spending extra effort to investigate the properties of some fundamental examples from this special class. While some of these examples are seemingly natural extensions of their realfunction counterparts, we will see that there are some significant differences between the real and complex versions. We also consider functions as mappings and review some basic terminology for discussing how sets of points in the function’s domain are mapped to their corresponding image sets in the codomain. We summarize the rules for combining functions to make new functions, and we note the special circumstances under which a given function has an associated inverse function. Finally, we investigate the important process of taking the limit of a function in our various cases of interest, a process that will be used throughout our remaining development of analysis.
