Removable discontinuity

Definition

A removable discontinuity occurs when a function has a hole at a particular point, which can be 'fixed' by redefining the function at that point. This happens when the limit of the function as it approaches the point exists but is not equal to the function's value at that point.

5 Must Know Facts For Your Next Test

1. Removable discontinuities are characterized by holes in the graph of a function.
2. They occur when $\lim_{{x \to c}} f(x)$ exists, but $f(c)$ is either undefined or not equal to this limit.
3. To remove a discontinuity, redefine $f(c)$ to be equal to $\lim_{{x \to c}} f(x)$.
4. Removable discontinuities do not affect the overall behavior of a function significantly and are considered less severe than other types of discontinuities.
5. Identifying removable discontinuities often involves factoring and simplifying rational functions.

Review Questions

• What conditions must be met for a discontinuity to be considered removable?
• How can you 'fix' a removable discontinuity in a function?
• Why are removable discontinuities considered less severe than other types of discontinuities?

Related terms

Limit:

The value that a function approaches as the input approaches some value.

Continuous Function: A function without any breaks, jumps, or holes in its graph.

Jump Discontinuity: A type of discontinuity where the left-hand and right-hand limits exist but are not equal.