Removable discontinuity

Definition

A removable discontinuity occurs at a point in a function where the function is not defined or does not match the limit, but can be 'removed' by redefining the function at that point. This type of discontinuity highlights important aspects of continuity and integrability, as it indicates that while the function may have a gap or break, it could be made continuous by appropriately assigning a value to the discontinuous point.

5 Must Know Facts For Your Next Test

1. Removable discontinuities often arise in rational functions where both the numerator and denominator share a common factor, leading to an undefined expression at certain points.
2. To remove the discontinuity, one can define the function at the point of discontinuity to equal the limit of the function as it approaches that point.
3. Removable discontinuities do not affect the Riemann integrability of a function, as they can be ignored for integration purposes if they occur over a finite interval.
4. Identifying removable discontinuities often involves finding limits and analyzing whether a hole exists in the graph of the function at specific points.
5. In practical terms, when graphing functions, removable discontinuities appear as holes, indicating that while the value is missing, it could potentially be filled in to achieve continuity.

Review Questions

• How do you identify a removable discontinuity in a function?
  • To identify a removable discontinuity in a function, you look for points where the function is either undefined or does not equal the limit as it approaches that point. A common method involves factoring the function to find any common factors between the numerator and denominator. If these factors can be canceled out, it indicates that there is a removable discontinuity at that point, showing that redefining the function at this location could restore continuity.
• Discuss how removable discontinuities relate to the properties of continuous functions.
  • Removable discontinuities illustrate key properties of continuous functions because they highlight situations where continuity fails due to specific values being undefined. Continuous functions require that limits coincide with the function values at all points, meaning if there's a removable discontinuity, one can redefine the function at that point to restore continuity. This showcases how understanding discontinuities helps in characterizing what makes functions continuous.
• Evaluate how removing a discontinuity impacts the Riemann integrability of a function over an interval.
  • Removing a discontinuity can significantly enhance the Riemann integrability of a function over an interval by allowing for smoother transitions without gaps. Since removable discontinuities are isolated points, they do not contribute to issues with Riemann sums used in integration. Therefore, by redefining such points, one ensures that integrability conditions are met, ultimately leading to better-defined areas under curves and more accurate integration results.

Related terms

Continuous Function: A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point.

Limit: The value that a function approaches as the input approaches a certain point. Limits are essential in understanding the behavior of functions around points of discontinuity.

Piecewise Function: A function that is defined by different expressions or formulas depending on the input value. These functions can exhibit removable discontinuities when transitioning between pieces.