In this part we introduce real functions. We will look at their properties here, but we limit ourselves here only to things that do not require differentiation. Thus we cover domain, operations with functions including composition, inverse function, boundedness, symmetry, periodicity, continuity, and we briefly introduce uniform continuity. We make a survey of elementary functions and also look at some useful and some weird function. However, the core topic of this chapter is limits, we especially focus on their evaluation. All these topics are also covered in Methods Survey, Solved Problems and Exercises. We also briefly introduce implicit and parametric functions.
We will also introduce the notions of monotonicity, local and global extrema, and concavity, we look at their properties, but the treatment is mostly theoretical. We do look at monotonicity in Solved Problems and Exercises, but we only use the definition there. The preferred approach to these topics is via differentiation, so we leave the practical treatment to the chapter Derivatives, where in section Graphing function we collected all methods from Functions and Derivatives that help us sketch the graph of a function, in particular we discuss monotonicity and local extrema, concavity, and asymptotes there.
Two most important topics (apart from thorough understanding of the notion of a function) is the domain and limits. Methods Survey, Solved Probems and above all Exercises should help you along the way. Exercises for limits are sorted by difficulty.
Since working with functions requires a good understanding of the topology of the set of real numbers, we included one part where we briefly review the basic set theory, look at mappings, recall popular sets of numbers and look closer at the real line (neighborhoods, intervals etc).