# Ch 13: Saxon Calculus: Continuity as a Property of Functions

### About This Chapter

## How It Works:

- Find the lesson within this chapter that corresponds to what you're studying in the Continuity as a Property of Functions chapter of your textbook.
- Watch fun videos that cover the calculus concepts you need to learn or review.
- Complete the quiz after watching each video lesson to test your understanding.
- If you need additional help, rewatch the videos until you've mastered the material, or submit a question for one of our instructors.

## Chapter Topics

You'll learn all of the calculus topics covered in the textbook chapter, including:

- Understanding continuity and discontinuity in functions and graphs
- Identifying regions of continuity
- Balzano's theorem and the extreme value theorem
- How to use the critical number theorem
- Polar and interval continuity
- Definition, applications and examples of the intermediate value theorem

*Saxon is a registered trademark of Houghton Mifflin Harcourt, which is not affiliated with Study.com.*

### 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. 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.

### 5. Using the Critical Number Theorem

In this lesson, we will review the statement of the critical number theorem and give several typical examples of its application in locating maximum and minimum values of a function.

### 6. Understanding Point Continuity

Continuity means unbroken or uninterrupted. Some functions are continuous across all values of x, and others aren't. In this lesson, we will discuss point continuity in terms of graphs.

### 7. Understanding Interval Continuity

Continuity means contiguous or continuing. Graphs of functions are either continuous between two x-values boundaries or not. In this lesson, we will investigate how to determine if a function is continuous in an interval.

### 8. 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.

### 9. 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

Other chapters within the Saxon Calculus Homeschool: Online Textbook Help course

- Saxon Calculus: Real Numbers
- Saxon Calculus: Algebra
- Saxon Calculus: Algebra Theorems
- Saxon Calculus: Geometry
- Saxon Calculus: Logic
- Saxon Calculus: Trigonometry
- Saxon Calculus: Graphing Calculator
- Saxon Calculus: Basics of Functions
- Saxon Calculus: Graphing Functions & Equations
- Saxon Calculus: Analysis of Graphs
- Saxon Calculus: Limits of Functions
- Saxon Calculus: Asymptotic & Unbounded Behavior
- Saxon Calculus: Parametric, Polar & Vector Functions
- Saxon Calculus: Concept of the Derivative
- Saxon Calculus: Derivative at a Point
- Saxon Calculus: Derivative as a Function
- Saxon Calculus: Second Derivatives
- Saxon Calculus: Applications of the Derivative
- Saxon Calculus: Computation of Derivatives
- Saxon Calculus: Riemann Sums
- Saxon Calculus: Interpretations & Properties of Definite Integrals
- Saxon Calculus: Applications of Integrals
- Saxon Calculus: Fundamental Theorem of Calculus
- Saxon Calculus: Techniques of Antidifferentiation
- Saxon Calculus: Applications of Antidifferentiation
- Saxon Calculus: Numerical Approximation of Definite Integrals
- Saxon Calculus: Concept of Series
- Saxon Calculus: Series of Constants
- Saxon Calculus: Taylor Series