A detailed look at continuity in Calculus textbooks

The aim of this paper is to contribute to the ongoing discussion about the role of intuition and ambiguity in doing, teaching and learning mathematics. We start by discussing the ways in which limits and continuity are presented in Calculus textbooks, to illustrate some of the ambiguities, and to contrast them with precise and rigorous definitions and statements. Next, we offer a brief overview of the topological terms necessary to introduce the notion of a continuous function in the topological setting. We conclude by suggesting a way of presenting the concept of a continuous function appropriate for Calculus instruction, while still staying true to its topological roots. This paper is also a suggestion to mathematics instructors to consider modifying the way they introduce limits and continuity in their classrooms, by creating a balance (appropriate for their students!) between rigour on the one side and intuition and ambiguity on the other.