Removable discontinuity

Definition

A removable discontinuity occurs in a function at a point where the function is not defined but can be made continuous by redefining the function at that point. It often appears as a 'hole' in the graph of the function.

5 Must Know Facts For Your Next Test

1. Removable discontinuities occur when both the left-hand and right-hand limits exist and are equal, but the function value is either undefined or different.
2. They can often be identified in rational functions where common factors cancel out in the numerator and denominator.
3. To remove the discontinuity, redefine the function at that point to equal the limit as it approaches that point.
4. Graphically, removable discontinuities appear as holes in otherwise smooth curves of functions.
5. In polynomial and rational functions, they are typically found by factoring and simplifying expressions.

Review Questions

• What conditions must be met for a discontinuity to be classified as removable?
• How can you identify a removable discontinuity in the expression of a rational function?
• Describe how you would remove a discontinuity at $x = c$ for a given rational function.

Related terms

Rational Function: A function defined by the ratio of two polynomials.

$\text{Limit}$: $\text{The value that a function approaches as} \ x \ \text{approaches some value}.$

$\text{Asymptote}$: $\text{A line that a graph approaches but never touches or crosses.}$