Functions, Limits, and Continuity

Introduction and Summary

In this preliminary section we review some definitions and concepts related to functions, the main objects of study in calculus. In calculus we are only interested in functions that vary smoothly, with values that do not jump around haphazardly. As a first attempt to make this idea rigorous, we will define a property of functions called continuity.

A continuous function is one whose values do not jump too suddenly, so it is important to understand the behavior of a function "near" a point. The mathematical expression of this idea is called a limit.

In this section we will also discuss several examples of functions, and comment on how limits and continuity apply in these cases. The types of functions discussed here (e.g. polynomial, rational, trigonometric, power) will arise very frequently in the subsequent sections.