About This Chapter
When functions are continuous, you're able to trace them with your finger on a graph, and at no point will the line break. If you trace a function with your finger and have to lift it off the paper at any point, then you know the function is discontinuous.
In this chapter's series of video lessons on continuity, you'll watch as fighter jets chase alien UFOs on a graph to help you understand continuous and discontinuous functions. Throughout these lessons, you'll learn about the various types of discontinuities and figure out if the fighter jets will eventually catch up to the UFOs.
You'll look at the three different types of discontinuities: removable discontinuity, jump discontinuity and asymptotic discontinuity. Watch as a UFO stops at one point and immediately reappears at another point along the graph. It may jump, separate or launch to positive and negative infinity at a particular point, which illustrates the discontinuity types.
Then, when graphing discontinuous functions, you'll learn how they may go through regions of continuity by sticking to one unbroken path. You'll also learn the intermediate value theorem, which will show why the fighter jet, at a certain point, will cross paths with the UFO.
After watching each lesson, you'll be able to test your knowledge through a short, multiple-choice quiz. There's also a chapter-ending test to measure your mastery of continuity.
1. Continuity in a Function
Travel to space and explore the difference between continuous and discontinuous functions in this lesson. Learn how determining continuity is as easy as tracing a line.
2. Discontinuities in Functions and Graphs
In this lesson, we talk about the types of discontinuities that you commonly see in functions. In particular, learn how to identify point, jump and asymptotic discontinuities.
3. Regions of Continuity in a Function
Can Earth ever compete with extraterrestrial UFOs? In this lesson, you'll learn that not all functions are continuous, but most have regions where they are continuous. Discover how to define regions of continuity for functions that have discontinuities.
4. Intermediate Value Theorem: Definition
A UFO and a jet take off and ascend to 30,000 feet along discontinuous and continuous paths, respectively. In this lesson, learn about the intermediate value theorem and why the jet has to cross 15,000 feet.
5. Intermediate Value Theorem: Examples and Applications
Many problems in math don't require an exact solution. Some problems exist simply to find out if any solution exists. In this lesson, we'll learn how to use the intermediate value theorem to answer an age-old question.
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Other chapters within the Math 104: Calculus course
- Graphing and Functions
- Vectors in Calculus
- Geometry and Trigonometry
- How to Use a Scientific Calculator
- Rate of Change
- Calculating Derivatives and Derivative Rules
- Graphing Derivatives and L'Hopital's Rule
- Applications of Derivatives
- Area Under the Curve and Integrals
- Integration and Integration Techniques
- Integration Applications
- Differential Equations
- Studying for Math 104