Explanation

A discontinuity occurs where a function is not defined.

For rational functions, this happens when:
Denominator = 0

Two types:

1. Removable discontinuity (hole)

   • Factor cancels

   • Appears as a hole in the graph

2. Vertical asymptote

   • Factor does NOT cancel

   • Function grows infinitely

Example:
(x − 2)/(x − 2) → hole at x = 2
1/(x − 2) → vertical asymptote at x = 2

Quiz

1. What causes a discontinuity?

2. What is a hole?

3. What creates a vertical asymptote?

4. Does canceling a factor remove the restriction?

5. Is the function defined at a vertical asymptote?

Answer Key

1. Denominator equals zero

2. A removable discontinuity

3. A non-canceling factor

4. No

5. No