Continuity

Abstract

The goal of this chapter is to explore the limit of a function whose domain is a subset of $\mathbb{R}$ and to then use this idea to investigate the notion of continuity. In the first-year calculus sequence, these topics are usually covered at a quick pace in an effort to move on to more interesting constructs such as the derivative and the integral. However, as the derivative is defined as a functional limit, it makes sense to have a thorough understanding of limits before moving forward. Historically, the derivative and the integral came before the ideas of limits and continuity. Many of the developments of the seventeenth and eighteenth centuries were motivated by a desire to understand the behavior of series of functions, specifically Fourier series. There were, however, complications that arose, and, as the mathematics developed, the need for a more stable foundation became apparent. In order to better understand Fourier series, mathematicians were forced to consider integration of such objects, and this study eventually led, in the early nineteenth century, to the formal development of the modern concept of limits. We have already discussed one particular type of limit, that of a sequence, and the more general notion of a functional limit is an extension of those ideas.