Skip to main content

Section 1.4 Now are the discontinuities of our discontent

Worksheet
A4US 

Now are the discontinuities of our discontent
Made glorious summer by these points returned;                
And all the jumps that lour'd upon our function                
In the deep bosom of the ocean buried.

(Apologies to Shakespeare 1 )

This is a mathy rewrite of Glouster's speech from Richard III

In this activity, we will examine a bunch of different curves in the plane and think about what things they have in common and what are their differences. Our goal is to arrive at some categories and definitions that allow us to distinguish between different types of functional behavior, with a focus on the idea of continuity. In rough terms, we think of a continuous function as one where you can draw it completely without lifting your pen from the page once you start.

After each of the three steps, we will return from our groups to discuss the results with the whole class.

1.

Below is a collection of cards with sketches of curves on them. For the next five minutes, your group should brainstorm about the following questions. What are the features of the curves that you see in each example? Which of them have commonalities? What are differences between them?

2.
Write down the definition of continuity as best you can from memory. Take a couple of minutes to debate the fine points and come to a consensus.
3.
Between the last step and this one, we will review the "book" definition of countinuity. What commonalities and differences do the definitions have? Are they equivalent? Which of your curves satisfy the definition and which don’t? Are there any other types of curves you can draw that don’t satisfy the definition?