To begin this lesson, we will recap the Warm-Up question, which will introduce the key concepts in calculus: Continuity, which is a fundamental requirement for most calculus theorems, and the Intermediate Value Theorem. You have seen both of these in action in your previous classes, but now we will frame them in the language of calculus. Duration: 6:52

Print a copy of "Ross's Beautiful Function Worksheet" below, assuming you have appropriate technology and the required amount of ink and/or toner. Otherwise, work through the investigation on your own paper. Side 1 hosts the graph of a piecewise function we will call f(x). Side 2 has a table to fill out based on the graph from Side 1. For the last column, you are making a decision about the continuity of f(x) at each point x=a based on your intuitive definition of continuity. Your end goal is to use inductive reasoning to come up with 3 necessary conditions for the continuity of a function at a given point. Check your work with the "Beautiful Function Answers" document.

Let's debrief on our investigation and define the three necessary conditions for the continuity of a function at a particular point. It would be wise of you to embroider these into your bedspread and study them when the lights go low, assuming you can actually still see. Perhaps it would be even wiser to use Braille. Duration: 12:14

On Example 2a, we'll apply the definition of continuity to a rational function expressed as an equation. This will illustrate our first type of discontinuity: Removable. Duration: 6:52

On Example 2b, we will discover the other types of non-removable discontinuities: Jump and Infinite. Duration: 10:12

Remember one-side limits? Similarly, we can have one-sided continuity. If your function is continuous from the right of x=a, confoundedly the point actually exists on the left side of the graph. Duration: 6:40

Now that we have internalized continuity at a point on a function, let's broaden the definition to include continuity on an interval. Duration: 3:25

Take a look, here are some possible solutions to Example 3. Duration: 4:06

In this final video for Objective 1, we'll examine a number of properties of continuous functions and establish the connection between domain and continuity. Duration: 13:27