Removable Discontinuity

Definition

A removable discontinuity occurs when a function is not defined at a certain point, or its value at that point differs from the limit of the function as it approaches that point. This type of discontinuity can be 'removed' by redefining the function at that point so that it becomes continuous. In essence, if you can fill in the hole in the graph of the function, it's considered removable.

5 Must Know Facts For Your Next Test

1. Removable discontinuities are characterized by having limits that exist, but the function itself is either undefined or has a different value at that point.
2. If a function has a removable discontinuity, it can often be rewritten as a new function that is continuous at that point by assigning it the limit value.
3. Graphically, removable discontinuities appear as holes in the graph of the function.
4. To determine if a discontinuity is removable, check if both one-sided limits exist and are equal to each other and to the value of the function where it is defined.
5. Common examples of functions with removable discontinuities include rational functions where factors can be canceled out.

Review Questions

• How can you identify a removable discontinuity in a function?
  • To identify a removable discontinuity in a function, you need to check if there is a point where the function is undefined or has a different value than its limit. This involves calculating the limits from both sides of that point and seeing if they exist and are equal. If they are equal but differ from the actual function value at that point, you've found a removable discontinuity.
• What steps would you take to redefine a function at a point with a removable discontinuity to make it continuous?
  • To redefine a function at a point with a removable discontinuity, first identify the limit of the function as it approaches that point from both sides. Then, assign this limit value to the function at that point. By doing this, you'll eliminate the hole in the graph and create a continuous function at that location.
• Evaluate how understanding removable discontinuities enhances your comprehension of overall continuity in functions.
  • Understanding removable discontinuities enhances comprehension of overall continuity because it shows how minor adjustments can lead to significant changes in a function's behavior. By recognizing that some discontinuities can be addressed through simple redefinitions, you gain insight into more complex behaviors of functions. This knowledge aids in analyzing functions more thoroughly, allowing you to better understand when and why functions behave continuously or discontinuously across their domains.

Related terms

Continuous Function:

A function is continuous at a point if the limit of the function as it approaches that point equals the value of the function at that point.

Limit: The limit of a function at a particular point describes the value that the function approaches as the input approaches that point.

Discontinuity: Discontinuity refers to a point in the domain of a function where the function is not continuous, which can include removable, jump, and infinite discontinuities.