Discontinuous Functions: Properties & Examples Video

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  • 0:00 What Is Discontinuous…
  • 0:54 Properties
  • 1:25 Removable Discontinuities
  • 2:10 Jump Discontinuities
  • 2:45 Asymptotic Discontinuities
  • 3:10 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

In this lesson, you will learn about what makes a function discontinuous. You'll also explore different types of discontinuity, look at properties of discontinuity, and better understand the concept through examples.

What Is a Discontinuous Function?

You may have already heard of a continuous function, which is a function on a graph that is a continuous curve. When you put your pencil down to draw it, you never lift your pencil up until the function is complete.

A discontinuous function is the opposite. It is a function that is not a continuous curve, meaning that it has points that are isolated from each other on a graph. When you put your pencil down to draw a discontinuous function, you must lift your pencil up at least one point before it is complete.

If you ever see a function with a break of any kind in it, then you know that function is discontinuous. In the function we have here, you can see how the function keeps going with a break.

The discontinuous function stops where x equals 1 and y equals 2, and picks up again where x equals 1 and y equals 4.

Properties

There are some properties that are specific to discontinuous functions, and two are particularly important:

First, the function always breaks off at a certain point or multiple points. As we have already discussed, discontinuous functions have points where the graph just stops and picks up somewhere else.

Second, the limit of the function at a point of discontinuity is undefined for most discontinuous functions, but not in all cases. The limit can be defined but is still considered discontinuous.

Now, let's explore some of the common types of discontinuous functions.

Removable Discontinuities

One type of discontinuity is called a removable discontinuity, or a hole. It is called removable because the point can be redefined to make the function continuous by matching the value at that point with the rest of the function.

When graphed, a removable discontinuity, or a hole, is just a missing value in the function. Everything else looks like a continuous graph. If we define that missing point, we will have removed the discontinuity.

The removable discontinuity is noted on the graph by a little circle at the point of discontinuity. Do you see how if we define that particular point to be the same as the function at that point, we will have removed the discontinuity?

Jump Discontinuities

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