Continuity

Emily Davis
5 min read
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Study Guide Overview
This guide covers removable discontinuities, focusing on their definition, identification, and removal. It explains how to determine if a discontinuity is removable by evaluating limits. The guide provides worked examples, practice questions, and exam strategies for handling these types of problems. Key terms include continuous function, limit, and removable discontinuity.
Study Notes on Removable Discontinuity
Table of Contents
- Introduction to Removable Discontinuity
- Definition and Concept
- Identifying Removable Discontinuities
- Removing Removable Discontinuities
- Worked Example
- Practice Questions
- Glossary
- Summary and Key Takeaways
- Exam Strategy
Introduction to Removable Discontinuity
Understanding removable discontinuities is crucial for mastering continuity in functions, a fundamental topic in calculus. This guide will help you identify and remove these discontinuities, ensuring a smooth learning experience.
Definition and Concept
What is a Removable Discontinuity?
A removable discontinuity is a type of discontinuity in a function that can be "removed" to make the function continuous over an interval that includes the discontinuity.
A function is continuous at the point if:
Characteristics of Removable Discontinuity
A removable discontinuity is essentially a "hole" in the function. It is a point where:
- exists,
- but does not exist.
Identifying Removable Discontinuities
To identify a removable discontinuity:
- Check if exists.
- Verify that does not exist or is not equal to the limit.
Students often confuse removable discontinuities with non-removable ones. Ensure you check both the existence of the limit and the function value at the point.
Removing Removable Discontinuities
How to Remove a Removable Discontinuity
To remove a removable discontinuity, you need to redefine the function at the point of discontinuity:
- Define to be equal to .
- This redefinition makes the function continuous at that point.
Worked Example
Let be the function defined by .
(a) Explain why is not continuous at .
Answer:
At :
Since is not defined, is not continuous at .
(b) Explain how the discontinuity at can be removed.
Answer:
First, find the limit as approaches 0 by factorizing and simplifying:
Then,
The limit exists, so the discontinuity at is removable. Redefine the function so that:
Thus, the redefined function is:
This removes the discontinuity.
Practice Questions
Practice Question
1. Identify the removable discontinuities in the following functions and redefine them to remove the discontinuities:
(a)
(b)
(c)
2. Determine if the following function has a removable discontinuity at :
3. Explain why is not continuous at and how to remove the discontinuity.
Glossary
- Removable Discontinuity: A discontinuity that can be "removed" by redefining the function at the point of discontinuity.
- Limit: The value that a function approaches as the input approaches some value.
- Continuous Function: A function without any breaks, jumps, or holes in its graph.
Summary and Key Takeaways
Key Points
- A removable discontinuity is a "hole" in a function where the limit exists, but the function value does not.
- To remove a removable discontinuity, redefine the function at the discontinuous point to equal the limit.
- Understanding limits and continuity is crucial for identifying and removing removable discontinuities.
Key Takeaways
- Always check the existence of the limit and the function value at the point of discontinuity.
- Redefine the function to make it continuous at the point of discontinuity.
Exam Strategy
When facing questions on continuity, always verify the existence of the function value and the limit separately.
If a function is not continuous at a point, check if the discontinuity is removable by finding the limit.
Practice identifying and removing removable discontinuities with different types of functions to build confidence.
Good luck with your studies, and remember, practice makes perfect!

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