Removable discontinuity

Definition

A removable discontinuity occurs in a function when there is a point where the function is not defined, but can be made continuous by appropriately defining or redefining the function at that point. This type of discontinuity often arises when a rational function has a common factor in both the numerator and denominator, leading to a hole in the graph at that specific x-value.

5 Must Know Facts For Your Next Test

1. Removable discontinuities can typically be identified by factoring the expression and canceling common terms to see if the limit exists at that point.
2. When graphing functions with removable discontinuities, there will be an open circle at the point of discontinuity to indicate it is not included in the function's values.
3. These discontinuities indicate that the limit of the function exists as you approach that x-value from either side, even though the function itself may not be defined there.
4. To remove a removable discontinuity, one can redefine the function at that specific x-value to equal the limit of the function as x approaches that value.
5. Identifying and addressing removable discontinuities is crucial when performing curve sketching to accurately represent the behavior of the function.

Review Questions

• How can you identify a removable discontinuity in a rational function?
  • To identify a removable discontinuity in a rational function, start by factoring both the numerator and denominator. Look for common factors; if you find one, it indicates a potential removable discontinuity at that x-value. After canceling these factors, check if the limit exists as you approach this x-value from both sides. If it does exist and the function is undefined at that point, then you have found a removable discontinuity.
• What are the implications of having a removable discontinuity on the graph of a function?
  • Having a removable discontinuity on a graph implies that while there is an undefined point (a hole) at that specific x-value, it doesn’t necessarily disrupt the overall continuity of the function elsewhere. It shows that although there is a gap in values, one could define or redefine the function at that point to fill in the hole, thereby creating a continuous function. Understanding this allows for better representation and sketching of functions with such characteristics.
• Evaluate how recognizing removable discontinuities influences curve sketching and understanding limits.
  • Recognizing removable discontinuities greatly influences curve sketching by indicating where to place open circles on graphs to show points that are excluded from the function's domain. It also enhances understanding of limits since it demonstrates situations where limits can exist even when a function doesn't take on certain values. This knowledge helps in accurately depicting how a graph behaves near these points and ensures that calculus concepts like continuity and differentiability are correctly applied during analysis.

Related terms

Continuous Function:

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

Non-removable Discontinuity: This type of discontinuity occurs when the limit does not equal the function's value at that point and cannot be 'fixed' by redefining the function.

Asymptote:

A line that a graph approaches but never touches, which often indicates a non-removable discontinuity in the context of rational functions.