AP Calculus AB and BC glossary

Removable discontinuity

Also called: Hole, Point discontinuity

A removable discontinuity is a point where the limit exists but the function either is not defined there or is defined to be a different value. It appears as a hole in the graph, and redefining the single point would make the function continuous.

These come from a factor that cancels. In $f(x) = \frac{x^2 - 4}{x - 2}$ , the numerator factors as $(x-2)(x+2)$ , so the function equals $x + 2$ everywhere except $x = 2$ , where it is undefined. The limit there is 4, so the hole sits at the point $(2, 4)$ .

The word removable is literal: define $f(2) = 4$ and the discontinuity is gone. That is impossible for a jump or an infinite discontinuity, which is what distinguishes them.

The mistake

Cancelling the factor and forgetting to exclude the point. The simplified expression $x + 2$ is a different function from the original unless you state that $x \ne 2$ .

Appears in: Unit 1: Limits and Continuity