About This Chapter
Overview of Function Continuity - Chapter Summary
Become familiar with the overview of function continuity with the help of this chapter. The lessons can help you to further your understanding of concepts such as continuity and discontinuity in a function and regions of continuity in a function. Our professional experts will also cover how Rolle's Theorem is related to the Mean Value Theorem. Once you finish this chapter, you should be able to:
- Understand continuity and discontinuity in function and graphs
- Analyze regions of continuity in a function
- Describe the intermediate value theorem and give examples
- Explain the mean value theorem
- Compare Rolle's theorem to the mean value theorem
- Clarify the extreme value theorem and bolzano's theorem
Our short and engaging videos were created to help you understand these calculus concepts quickly and easily. If you struggle to understand a concept, use the timeline tags within the videos to go back to a certain part of the video to review. You can request help from an expert using the Dashboard if needed.
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.
6. What is the Mean Value Theorem?
Three people set off on a car trip. They all start at the same time and end at the same time. Learn what calculus says about how fast they traveled along the way as you study the Mean Value Theorem in this lesson.
7. Rolle's Theorem: A Special Case of the Mean Value Theorem
Super C, the human cannonball, is shot into the air at 35 mph, but his average vertical velocity is zero. In this lesson, you will use Rolle's theorem to explain what this means about Super C's flight.
8. Extreme Value Theorem & Bolzano's Theorem
The extreme value theorem and Bolzano's theorem are two very useful theorems that you can use to help you find solutions as well as maximums and minimums of a function. Learn how to use them in this lesson.
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Other chapters within the AP Calculus AB & BC: Exam Prep course
- Graph Basics
- The Basics of Functions
- How to Graph Functions
- Overview of Limits of Functions
- Understanding Exponentials & Logarithms
- Using Exponents and Polynomials
- Parametric, Polar and Vector Functions
- Overview of Properties of Derivatives
- The Derivative at a Point
- The Derivative as a Function
- Second Derivatives
- Derivative Applications
- Finding Derivatives
- Properties of Definite Integrals
- Applications of Integrals
- Using the Fundamental Theorem of Calculus
- Applying Integration Techniques
- Approximation of Definite Integrals
- Understanding Sequences & Series
- Series of Constants
- Taylor Series
- Using a Scientific Calculator for Calculus
- AP Calculus AB & BC Flashcards