About This Chapter
Continuity of Functions - Chapter Summary and Learning Objectives
Continuity is a particular characteristic of functions that deals with a function's output. Continuity means that a function will approach a certain output at a certain input, and it will equal that same output at that same input. This chapter examines the formal mathematical expressions of continuity. It also explores the different indicators and properties of continuous and discontinuous functions. The lessons in this chapter are short, about 5-10 minutes each, and come with lesson transcripts and quizzes to help you quickly learn the material. By the end of the chapter, you will have mastered:
- The definition of a continuous function
- How to discover discontinuous functions
- Graphical representations of continuous and discontinuous functions
- The Intermediate Value Theorem and its applications
- How to define a function's regions of continuity
|Continuity in a Function||Define continuity and its corresponding mathematical expressions.|
|Discontinuities in Functions and Graphs||Learn to spot discontinuous functions. Explore representations of discontinuity in a Cartesian coordinate system.|
|Regions of Continuity in a Function||Consider intervals of functions and explore defining a functions continuous regions.|
|Intermediate Value Theorem||Master the formal definition of the Intermediate Value Theorem.|
|Intermediate Value Theorem: Examples and Applications||Practice using the Intermediate Value Theorem.|
|Continuous Functions Theorems||Review various theorems that detail properties of continuous functions.|
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 AB: Exam Prep course
- Graph Basics
- The Basics of Functions
- How to Graph Functions
- Limits of Functions
- Understanding Exponentials & Logarithms
- Using Exponents and Polynomials
- Properties of Derivatives
- The Derivative at a Point
- The Derivative as a Function
- Second Derivatives
- Using Derivatives
- Computing Derivatives
- Properties of Definite Integrals
- Applications of Integrals
- Using the Fundamental Theorem of Calculus
- Understanding & Applying Integration Techniques
- Approximating Definite Integrals
- Using a Scientific Calculator for Calculus
- About the AP Calculus Exam
- AP Calculus AB Flashcards