Removable discontinuity

Definition

A removable discontinuity occurs at a point in a function where the function is not defined, but can be made continuous by appropriately redefining the function at that point. This type of discontinuity arises when the limit of the function exists at that point, but the function itself does not take on that limit value. Recognizing and addressing removable discontinuities is crucial for understanding the behavior of functions and their continuity.

5 Must Know Facts For Your Next Test

1. Removable discontinuities can often be identified by factoring and canceling terms in rational functions.
2. To determine if a discontinuity is removable, you can check if the limit exists as you approach that point from both sides.
3. When redefined correctly, the limit of a function at a point with a removable discontinuity can equal the value of the function, thereby making it continuous.
4. Common examples of removable discontinuities include functions like f(x) = (x^2 - 1)/(x - 1), which has a hole at x = 1.
5. In calculus, handling removable discontinuities often involves using limits to find the correct value to assign to the function at the point of discontinuity.

Review Questions

• How can you identify a removable discontinuity in a function?
  • A removable discontinuity can be identified by checking if the limit of the function exists at the point where it is not defined. If you can factor and simplify the function, and if the limit from both sides approaches the same value, then it's likely that there's a removable discontinuity. For instance, in rational functions, if a term cancels out leading to an undefined value at a specific point, that's often where you'll find this type of discontinuity.
• Explain how to resolve a removable discontinuity in a function.
  • To resolve a removable discontinuity, you first determine the limit of the function as it approaches the point of discontinuity. Then, redefine the function at that specific point to match this limit. By doing this, you effectively 'fill in the hole' in the graph, ensuring that the function becomes continuous at that location. This process helps clarify how values behave near points where they were previously undefined.
• Compare and contrast removable and non-removable discontinuities, providing examples of each.
  • Removable and non-removable discontinuities differ significantly in their nature and resolution. A removable discontinuity occurs when a limit exists at a point but the function is not defined there; for example, f(x) = (x^2 - 1)/(x - 1) has a removable discontinuity at x = 1. In contrast, non-removable discontinuities arise when limits do not exist due to vertical asymptotes or jumps in values; for instance, f(x) = 1/(x - 2) has a non-removable discontinuity at x = 2 because it tends toward infinity. Understanding these differences is crucial for analyzing function behavior.

Related terms

limit: The value that a function approaches as the input approaches a certain point.

continuity: A property of a function where it is uninterrupted and has no breaks or jumps in its graph.

non-removable discontinuity: A type of discontinuity that cannot be eliminated by redefining the function at a particular point, often due to an infinite limit or a jump.