Intermediate Real Analysis

Abstract

In this chapter we will review, by way of problems, the hierarchy of definitions and results concerning continuous, differentiable, and integrable functions. We will build on the reader’s understanding of limits to review the most important definitions (continuity in Section 6.1, differentiability in Section 6.3, and integrability in Section 6.8). We will also call attention to the most important properties of these classes of functions. It is useful to know, for example, that if a problem involves a continuous function, then we might be able to apply the intermediate-value theorem or the extreme-value theorem; or again, if the problem involves a differentiable function, we might expect to apply the mean-value theorem. Examples of these applications are included in this chapter, as well as applications of L’Hopital’s rule and the fundamental theorem of calculus.