Continuity in the Calculus I Pathway

Continuity is a property of functions with several important implications. The conclusions of Rolle’s theorem, the mean value theorem, the intermediate value theorem, the extreme value theorem, and the fundamental theorem of calculus all require a function to be continuous on a closed interval. Continuity is also a necessary condition for integrability and for the algebraic properties of definite integrals. It is essential that students understand the relationship between continuity and differentiability, and leverage their understanding of continuity to make strategic inferences about function behavior.

Faculty at the MIP Calculus I Initiation Workshop in August, 2019 recommended that instructional resources developed by CoRDs and ARCs addressing Continuity for success in the Oklahoma Calculus I Pathway should:

1. Connect/distinguish continuity at a point with the limit of a function at that point.

2. Understanding the relationship between differentiability and continuity, namely that differentiability implies continuity.

3. Relate points on non-differentiability (e.g., “corners,” “cusps,” “vertical tangents”) to particular discontinuities of the derivative function.

4. Appreciate the role of continuity as a requirement in a variety of important theorems (e.g., intermediate value theorem, mean value theorem, extreme value theorem, integrability, the fundamental theorem of calculus).

5. Understand the relationship between one-sided limits, two-sided limits, the value of a function at a point, and continuity.

6. Interpret various types of discontinuity (e.g., “jump,” “removable,” “oscillating,” “infinite”) in terms of limit expressions.

7. Connect the limit definition of continuity to a function’s graph and to intuitive notions of continuity (i.e., “you can sketch the graph without lifting your pencil”).

8. Understand of continuity as an extension of limits.

9. Understanding the role of continuity in differentiability conditions.

Addressing Components of Inquiry

Participants of the MIP Workshop on Calculus I suggested the following ways resources about Continuity could address the three MIP components of mathematical inquiry:

Active Learning: An instructor might engage students in active learning to support their understanding of continuity and its implications by presenting tasks that require students to find the value of a parameter to make a piecewise defined function continuous where at least one of the expressions of the function involves computing a limit using a method other than direct evaluation. Such tasks require students to apply their understanding that the limit of one expression must be equal to the value of the other at a particular input. Other tasks that might support students’ active learning include asking them to sketch a graph of piecewise function given information about its one-sided limits and value at a point. Crucially, these tasks require students to coordinate the limiting value of a function with its value to assess whether the function is continuous at a particular point.

Meaningful Applications: Students’ understanding of continuity can be reinforced and extended by engaging in applied tasks that require them to use continuity to compute the limit of a function and to interpret computational strategies for limit evaluation as an instance of leveraging the definition of continuity to evaluate a limit. Additionally, applied contexts can support students’ recognition of the importance of continuity as a hypothesis in the intermediate value theorem, mean value theorem, and extreme value theorem by exploring counterexamples.

Academic Success Skills: Supporting students’ understanding of why the conclusions of important theorems in calculus depend on a function being continuous on a closed interval enables students to interpret these conclusions as intuitive implications of continuity, not as a list of facts to be memorized and applied to solve routine exercises. Interpreting the conclusions of these theorems as intuitive implications challenges the common assumption that mathematical proficiency is based principally on one’s ability to efficiently recall declarative knowledge—a perspective that increases students’ uncertainty as to whether they can successfully participate in mathematics. Constructing meaning for continuity and its implications allows students to recognize that a small number of essential ways of reasoning are sufficient for engaging productively in a variety of tasks, and if students have experienced their capacity to engage in these ways of reasoning in several mathematical contexts, then they are more likely to appraise task demands as manageable, thus encouraging perseverance in problem solving and a growth mindset about mathematical ability.

References

Jayakody, G., & Zazkis, R. (2015). Continuous problem of function continuity. For the Learning of Mathematics, 35(1), 8-14.

Maharajh, N., Brijlall, D., & Govender, N. (2008). Preservice mathematics students’ notions of the concept definition of continuity in calculus through collaborative instructional design worksheets. African Journal of Research in Mathematics, Science and Technology Education, 12(1), 93-106.

Patenaude, R. E. (2013). The use of applets for developing understanding in mathematics: A case study using Maplets for Calculus with continuity concepts (Doctoral dissertation, University of South Carolina).

Tall, D., & Katz, M. (2014). A cognitive analysis of Cauchy’s conceptions of function, continuity, limit and infinitesimal, with implications for teaching the calculus. Educational Studies in Mathematics, 86(1), 97-124.