A detailed look at continuity in Calculus textbooks

Abstract

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.

Keywords:

• Limits
• continuity
• Calculus textbooks
• definitions
• interpretation of mathematics

Acknowledgments

We sincerely thank the reviewers, whose suggestions made us think and helped us to improve the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 A subset I of real numbers is a closed and bounded interval if there are real numbers α and β, $\alpha <\beta$, such $I=\{x\in \mathbb{R}:\alpha \le x\le \beta \}$. We write $I=[\alpha ,\beta ]$.

2 A subset I of real numbers is an open interval if there are real numbers α and β, $\alpha <\beta$, such that $I=\{x\in \mathbb{R}:\alpha <x<\beta \}$ or if there is a real number γ such that $I=\{x\in \mathbb{R}:x<\gamma \},$ or $I=\{x\in \mathbb{R}:x>\gamma \},$ or if $I=\mathbb{R}$. We write, respectively, $I=(a,b),$ or $I=(-\mathrm{\infty },\gamma ),$ or $I=(\gamma ,\mathrm{\infty }),$ or $I=(-\mathrm{\infty },\mathrm{\infty })$.

3 We remind the reader that a definition in mathematics is always an ‘if and only if’ statement.

4 The Intermediate Value Theorem is commonly introduced in the same section/lecture as the definition of the continuity of a function.

5 We follow the terminology and notation in André (2023).

6 In all Calculus textbooks we have examined, the definition of a function continuous on a set uses the definition of a function continuous at a point.

7 This union can consist of a single interval. We emphasise that the reason why we said ‘union of intervals’ is to keep in mind that, in the context of university calculus, we do not consider the most general domains. As stated in footnotes 1 and 2, the endpoints of an interval are two distinct numbers, thus a single real number is not viewed as an interval.

8 Stewart, Clegg, and Watson use the term ‘points of discontinuity’. This terminology is common to all Calculus textbooks that we have encountered. By opting for the term singularity rather than a point of discontinuity we emphasise the fact that the continuity is defined only at points in the domain of a function.