Definition of Removable Discontinuity

Oct 13, 2025 by miodeya.com 114 views

Removable discontinuity is a fundamental concept in mathematics, particularly in calculus and real analysis. It is a type of discontinuity that occurs when a function is not continuous at a specific point, but the function can be made continuous by redefining the value of the function at that point. In this article, we will delve into the concept of removable discontinuity, its definition, and examples to illustrate this concept.

What is a Removable Discontinuity?

A removable discontinuity is a type of discontinuity that occurs when a function is not continuous at a specific point, but the function can be made continuous by redefining the value of the function at that point. This means that the function is not defined or is not continuous at the point of discontinuity, but the function can be made continuous by assigning a specific value to the function at that point.

Definition of Removable Discontinuity

A function $$f$$ is said to have a removable discontinuity at $$a$$ if:

$f$$ is either not defined or not continuous at $$a$$.$
$f(a)$$ could either be defined or redefined so that the new function is continuous at $$a$$.$

In other words, a removable discontinuity occurs when a function is not continuous at a point, but the function can be made continuous by redefining the value of the function at that point.

Examples of Removable Discontinuity

Let's consider some examples to illustrate the concept of removable discontinuity.

Example 1: A Simple Function

Consider the function $$f(x) = \frac{x^2 - 4}{x - 2}$$. This function is not defined at $$x = 2$$ because it results in division by zero. However, we can redefine the function at $$x = 2$$ by assigning a specific value to the function at that point. For example, we can define $$f(2) = 4$$, which makes the function continuous at $$x = 2$$. This is an example of a removable discontinuity.

Example 2: A Piecewise Function

Consider the piecewise function $$f(x) = \begin{cases} x^2 & \text{if } x < 2 \ 4 & \text{if } x = 2 \ x + 1 & \text{if } x > 2 \end{cases}$$. This function is not continuous at $$x = 2$$ because the left-hand limit and right-hand limit do not agree. However, we can redefine the function at $$x = 2$$ by assigning a specific value to the function at that point. For example, we can define $$f(2) = 4$$, which makes the function continuous at $$x = 2$$. This is an example of a removable discontinuity.

Properties of Removable Discontinuity

Removable discontinuity has several properties that are worth noting.

Property 1: Removable Discontinuity is a Type of Discontinuity

Removable discontinuity is a type of discontinuity that occurs when a function is not continuous at a specific point, but the function can be made continuous by redefining the value of the function at that point.

Property 2: Removable Discontinuity is Not a Gap

Removable discontinuity is not a gap in the function, but rather a point where the function is not continuous.

Property 3: Removable Discontinuity Can be Removed

Removable discontinuity can be removed by redefining the value of the function at the point of discontinuity.

Conclusion

In conclusion, removable discontinuity is a fundamental concept in mathematics that occurs when a function is not continuous at a specific point, but the function can be made continuous by redefining the value of the function at that point. This concept is essential in calculus and real analysis, and it has several properties that are worth noting. By understanding removable discontinuity, we can better analyze and understand the behavior of functions.

References

[1] Rudin, W. (1976). Principles of Mathematical Analysis. McGraw-Hill.
[2] Spivak, M. (1965). Calculus. W.A. Benjamin.
[3] Apostol, T. M. (1974). Calculus. Waltham, MA: Blaisdell.

Q: What is the difference between a removable discontinuity and a non-removable discontinuity?

A: A removable discontinuity is a type of discontinuity that occurs when a function is not continuous at a specific point, but the function can be made continuous by redefining the value of the function at that point. A non-removable discontinuity, on the other hand, is a type of discontinuity that occurs when a function is not continuous at a specific point, and the function cannot be made continuous by redefining the value of the function at that point.

Q: How do I determine if a function has a removable discontinuity?

A: To determine if a function has a removable discontinuity, you need to check if the function is not continuous at a specific point, but the function can be made continuous by redefining the value of the function at that point. You can do this by checking the left-hand limit and right-hand limit of the function at the point of discontinuity. If the left-hand limit and right-hand limit agree, then the function has a removable discontinuity.

Q: Can a removable discontinuity be removed by simply redefining the value of the function at the point of discontinuity?

A: Yes, a removable discontinuity can be removed by simply redefining the value of the function at the point of discontinuity. However, it's worth noting that redefining the value of the function at the point of discontinuity may change the behavior of the function in other ways.

Q: Are all removable discontinuities removable by redefining the value of the function at the point of discontinuity?

A: No, not all removable discontinuities can be removed by redefining the value of the function at the point of discontinuity. Some removable discontinuities may require more complex redefinitions of the function to remove.

Q: Can a removable discontinuity occur at a point where the function is not defined?

A: Yes, a removable discontinuity can occur at a point where the function is not defined. In this case, the function can be made continuous by redefining the value of the function at that point.

Q: Can a removable discontinuity occur at a point where the function is defined but not continuous?

A: Yes, a removable discontinuity can occur at a point where the function is defined but not continuous. In this case, the function can be made continuous by redefining the value of the function at that point.

Q: How do I find the value of a removable discontinuity?

A: To find the value of a removable discontinuity, you need to reevaluate the function at the point of discontinuity. You can do this by using the definition of the function and the properties of limits.

Q: Can a removable discontinuity be removed by using a different definition of the function?

A: Yes, a removable discontinuity can be removed by using a different definition of the function. However, it's worth noting that using a different definition of the function may change the behavior of the function in other ways.

Q: Are removable discontinuities always removable by redefining the value of the function at the point of discontinuity?

A: No, removable discontinuities are not always removable by redefining the value of the function at the point of discontinuity. Some removable discontinuities may require more complex redefinitions of the function to remove.

Q: Can a removable discontinuity occur at a point where the function is not differentiable?

A: Yes, a removable discontinuity can occur at a point where the function is not differentiable. In this case, the function can be made continuous by redefining the value of the function at that point.

Q: Can a removable discontinuity occur at a point where the function is not integrable?

A: Yes, a removable discontinuity can occur at a point where the function is not integrable. In this case, the function can be made continuous by redefining the value of the function at that point.

Conclusion

In conclusion, removable discontinuity is a fundamental concept in mathematics that occurs when a function is not continuous at a specific point, but the function can be made continuous by redefining the value of the function at that point. By understanding removable discontinuity, we can better analyze and understand the behavior of functions.

References

[1] Rudin, W. (1976). Principles of Mathematical Analysis. McGraw-Hill.
[2] Spivak, M. (1965). Calculus. W.A. Benjamin.
[3] Apostol, T. M. (1974). Calculus. Waltham, MA: Blaisdell.