Removable Discontinuity

Definition

A removable discontinuity occurs in a function when there is a point at which the function is not defined or does not have a limit, but can be 'fixed' by redefining the function at that point. This type of discontinuity is often characterized by a hole in the graph of the function, which indicates that the limit exists at that point but does not equal the function's value there. Recognizing removable discontinuities is crucial for understanding limits and continuity, as they indicate where a function can be made continuous by a simple adjustment.

5 Must Know Facts For Your Next Test

1. A removable discontinuity often appears in rational functions where a factor cancels out, resulting in a hole in the graph.
2. To determine if a discontinuity is removable, check if the limit exists at that point and if it is different from the actual value of the function.
3. Removable discontinuities can often be resolved by redefining the function to match the limit value at that point.
4. Graphically, a removable discontinuity looks like an open circle on the graph, showing where the function does not exist despite having a limit.
5. Recognizing and addressing removable discontinuities is important when finding limits and analyzing continuity in functions.

Review Questions

• How can you identify a removable discontinuity in a given function?
  • To identify a removable discontinuity, look for points in the function where it is undefined but has a finite limit. You can often find these points in rational functions where a term cancels out. By calculating limits approaching that point from both sides, you can confirm if they are equal. If they are equal but do not match the actual value of the function at that point, then you have a removable discontinuity.
• Discuss how you can resolve a removable discontinuity to make a function continuous.
  • To resolve a removable discontinuity and make the function continuous, you redefine the function at that specific point to match its limit. For example, if you find that the limit as x approaches a certain value equals L but the function is undefined at that value, you can define or redefine the function to be L at that point. This effectively 'fills in' the hole in the graph and ensures that both sides of the limit equal the same value, restoring continuity.
• Evaluate how understanding removable discontinuities impacts your approach to solving limits and ensuring continuity in more complex functions.
  • Understanding removable discontinuities is essential because it allows you to simplify complex functions and find limits more effectively. When analyzing piecewise or rational functions, recognizing these points helps avoid unnecessary complications during calculations. By addressing removable discontinuities first, you can streamline your work on limits and ensure continuity throughout your analysis. This foundational knowledge can lead to clearer insights into overall behavior and properties of functions as you progress to more intricate mathematical problems.

Related terms

Continuous Function: A function is continuous if there are no breaks, jumps, or holes in its graph over its entire domain, meaning the limit at every point equals the function's value.

Limit:

The limit of a function describes the value that the function approaches as the input approaches a particular point, even if the function is not defined at that point.

Non-removable Discontinuity: A non-removable discontinuity occurs when a function has a jump or infinite discontinuity, meaning it cannot be 'fixed' simply by redefining the function at a specific point.