Jump Discontinuities: Definition & Concept

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  • 0:00 What Is a Jump Discontinuity?
  • 0:31 What a Jump…
  • 2:01 Piecewise Functions
  • 3:16 Checking for Jump…
  • 4:53 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.

Learn why jump discontinuities are an interesting phenomenon in math and how you can identify functions that have them. Learn what they look like and how to check functions for their existence.

What Is a Jump Discontinuity?

When we have a function that jumps from one location to another at a certain point, we say we have a jump discontinuity. It's an interesting phenomenon because you can see the split in the function as it stops somewhere and picks up again somewhere else. The function can even behave differently after it picks up again. One certain thing is that the jump discontinuity makes the function not continuous. So, what does a jump discontinuity look like?

What a Jump Discontinuity Looks Like

Looking at a jump discontinuity, you'll see that the function just jumps from one location to another at a certain point or points. You can have more than one jump discontinuity in a given function. You can have anywhere from zero to hundreds. There is no limit. Let's look at a couple of examples of jump discontinuities to see a pattern of what they look like.

jump discontinuity

In this one, we see that at x = -2, the function stops at y = 1 and picks up again at y = 5. Because the function has jumped locations, this break is a jump discontinuity. Also, notice how the function behaves differently before and after the jump. Before the jump, the function is going downwards, but after the jump, the function starts going straight. This is perfectly acceptable for functions with jump discontinuities. Let's look at another example.

jump discontinuity

This one is a bit more interesting. Because all the breaks in the function are jumps, they are all jump discontinuities - and there are quite a few in this function. Between the jumps, we see that the function always behaves in a straight line, though in different locations. This is acceptable, too.

You may have noticed in these images that there are solid circles and open circles. These different types of circles do have different meanings. A circle of either type means that the function is stopping or beginning at that point. A solid circle means the function includes this value that it's sitting on. An open circle means the function is simply stopping or beginning at that point but does not include that point.

Piecewise Functions

Functions with jump discontinuities, when written out mathematically, are called piecewise functions because they are defined piece by piece. You'll see how the open and closed circles come into play with these functions. Let's look at a function now to see what a piecewise function looks like.

jump discontinuity

The formula tells us that for all x-values less than -1 but not including it, the function behaves as y = x + 1, and for all x-values greater than -1 including that number, the function behaves as y = x + 4. When graphing it out, you see that we have a different jump discontinuity at x=-1 because the function stops and begins again at different locations.

jump discontinuity

Did you notice how the function y = x + 1 does not include the point x = -1? That point is marked with an open circle to show that. The other point for the function y = x + 4, is marked with a closed circle to show that it does include the point x = -1.

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