1.13 Removing Discontinuities

Patching a hole

A removable discontinuity (a hole) can be removed by redefining f(c) to equal the limit. This often happens when a factor cancels in a rational expression.

Only removable ones

Only removable discontinuities can be fixed this way. Once patched, the function becomes continuous at that point. Jumps and infinite discontinuities cannot be removed.

Key terms

Removable discontinuity

A hole where the limit exists but ≠ f(c).

How to remove it

Redefine f(c) to equal the limit.

Canceled factor

Common source of a removable hole in rational functions.

Fixable vs. not

Only removable discontinuities can be patched.

Example

(x²−1)/(x−1) → define f(1) = 2.

Try an AP-style question

f(x) = (x² + x − 12)/(x − 3). What value of f(3) removes the discontinuity?

1. −4
2. 7
3. 3
4. 0