About This Chapter
Function Continuity - Chapter Summary and Learning Objectives
Our expert instructors will help you boost your knowledge of continuous function theorems such as the intermediate value and Bolzano's theorems. The brief quizzes after each lesson and the test at the end of the chapter are designed to give you an idea of where you strengths and weaknesses in this topic lie. The information covered in this chapter include:
- Definitions and examples of continuities and discontinuities in functions and graphs
- Examples and uses of the Intermediate Value Theorem
- Other continuous function theorems, including Extreme Value and Bolzano's
|Continuity in a Function||Here our instructors set out the rules for what qualifies as a continuous function (it is a function which does not have any breaks in its line, by the way).|
|Discontinuities in Functions and Graphs||This lesson describes several different types of discontinuities.|
|Regions of Continuity in a Function||In this video you'll learn about functions which have continuous sections between points of discontinuity.|
|Intermediate Value Theorem: Definition||Though it may go without saying, this lesson discusses the theorem which says that a line that is continuous between two points meets every value between those points at least once.|
|Intermediate Value Theorem: Examples and Applications||Our instructors walk you through a few examples of the intermediate value theorem here.|
|Extreme Value Theorem & Bolzano's Theorem||Learn about a few other theorems which work with continuous functions in this brief video.|
|Continuous Functions Theorems||Still more theorems working with continuous functions are described here.|
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. 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 BC: Exam Prep course
- Graph Basics
- The Basics of Functions
- How to Graph Functions
- Limits of Functions
- Understanding Exponentials & Logarithms
- Using Exponents and Polynomials
- Parametric, Polar and Vector Functions
- 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 BC Flashcards