Removable discontinuities

Definition

Removable discontinuities occur in a function when there is a point at which the function is not defined or does not match the expected value, but the limit exists at that point. These types of discontinuities can often be 'removed' by redefining the function at that specific point so that it becomes continuous. This is important because it allows for functions to be made continuous through adjustments, making them easier to analyze and work with.

5 Must Know Facts For Your Next Test

1. Removable discontinuities can usually be identified by a hole in the graph of the function at a certain x-value.
2. To 'remove' a removable discontinuity, you can redefine the function at that point to equal the limit of the function as it approaches that x-value.
3. Not all discontinuities are removable; if a limit does not exist at a point or goes to infinity, it is considered a non-removable discontinuity.
4. Removable discontinuities often appear in rational functions where factors in the numerator and denominator cancel out.
5. In calculus, recognizing and addressing removable discontinuities is crucial for evaluating limits and finding derivatives.

Review Questions

• How can identifying removable discontinuities help in analyzing functions?
  • Identifying removable discontinuities is key because it allows us to see where a function might be made continuous. When we find a hole in the graph of a function, we can often redefine it at that point to match the limit. This adjustment simplifies our analysis and helps us apply calculus concepts more effectively, like finding derivatives or integrals.
• Discuss how removable discontinuities differ from non-removable discontinuities and provide an example of each.
  • Removable discontinuities occur when a limit exists at a point but the function is not defined or does not match that limit, allowing for correction by redefining the function. An example is the function f(x) = (x^2 - 1)/(x - 1), which has a removable discontinuity at x = 1 since it simplifies to f(x) = x + 1 everywhere else. Non-removable discontinuities occur when limits do not exist or go to infinity, such as with f(x) = 1/(x - 2), which has a vertical asymptote at x = 2.
• Evaluate how removing a discontinuity affects the overall behavior of a function across its domain.
  • Removing a discontinuity can significantly improve the behavior of a function across its domain. By redefining the function at points where there were holes, we create a smoother transition without jumps or breaks. This continuity allows for better predictions of how the function behaves as inputs change and facilitates applying calculus techniques for analysis and solving equations related to that function.

Related terms

continuous function:

A function is considered continuous if there are no breaks, jumps, or holes in its graph, meaning it can be drawn without lifting a pencil.

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

point discontinuity: A type of discontinuity that occurs at a specific point in the domain of a function, where the function either does not exist or does not equal the limit at that point.