http://www.ck12.org Chapter 1. Functions and Graphs
Jump discontinuitiesoccur when a function has two ends that don’t meet even if the hole is filled in. In order to
satisfy the vertical line test and make sure the graph is truly that of a function, only one of the end points may be
filled. Below is an example of a function with a jump discontinuity.
Infinite discontinuitiesoccur when a function has a vertical asymptote on one or both sides. This is shown in the
graph of the function below atx=1.
Example A
Identify the discontinuity of the function algebraically and then graph the function.
f(x) =(x−^2 )((xx+− 12 )()x−^1 )
Solution:Since the factorx−1 is in both the numerator and the denominator, there is a removable discontinuity at
x=1.
When graphing the function, you should cancel the removable factor, graph like usual and then insert a hole in the
appropriate spot at the end.
Example B
Identify the discontinuity of the piecewise function graphically.