C. Types of Discontinuity

Types of Discontinuities

As seen in the video, there are two types of discontinuities: removable and non-removable discontinuities. And there are two types of non-removable discontinuities: jump and infinite discontinuities.

A removable discontinuity occurs when the graph of a function has a hole.

For example, consider the following function:

Notice that the set of factors x+1x+1can be removed or canceled to get the function f(x)=x+2. The graph will then resemble y =x+2, except there will be a hole at x =−1 to account for the removed factor x +1. The reason for the hole is that although the original function can be simplified, -1 must be excluded from the domain of the function. Thus, graph the line y =x+2 as usual, but remove the point at x =−1:

Removable discontinuities can be “filled in” if you make the function a piecewise function and define a part of the function at the point where the hole is. In the example above, to make f(x)continuous, you could redefine it as:

An infinite discontinuity occurs when a function has a vertical asymptote on one or both sides. This is shown in the graph of the function below at x=1:

Example 1

Identify the discontinuity of the function algebraically and then graph the function:

Solution:

The factor x −1 can be removed or cancelled in both the numerator and the denominator to result in f(x)=(x−2)(x+2). Because x−1 was cancelled, there is a removable discontinuity or hole at x=1.

The graph will resemble y(x−2)(x+2) with a hole at x=1x=1, so graph y=(x−2)(x+2) as usual and then insert a hole in the appropriate spot at the end:

Summary

There are two types of discontinuities: removable and non-removable. Then there are two types of non-removable discontinuities: jump or infinite discontinuities.

Removable discontinuities are also known as holes. They occur when factors can be algebraically removed or canceled from rational functions.

Jump discontinuities occur when a function has two ends that don’t meet, even if the hole is filled in at one of the ends.

Infinite discontinuities occur when a function has a vertical asymptote on one or both sides.