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.
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.
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.
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.
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.
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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Just because it has a piecewise function does not mean that it will have a jump discontinuity. Also, piecewise functions can have as many regions as they want to have. They are not limited to just two regions.
Checking for Jump Discontinuities
It's important to check for the existence of jump discontinuities even if you are given a piecewise function that looks like it has a jump discontinuity. How do you check for them? What you do is you plug in your x-value where the piecewise function changes to see if the different functions match each other or not.
Let's do an example to see how this works. We have a function like this:
Does this function have a jump discontinuity at x = 2? At first glance, it may look like that because the functions are different, and the function changes at x = 2. We can check by plugging in x = 2 into the functions. For x values less than 2, the function is y = x + 1. Plugging in x = 2, you get y = 2 + 1 = 3. Okay, that's good so far. Looking at the function again, you can see that for x-values greater than or equal to 2 the function is 3. So, at x = 2 for this part, then y = 3. Well, it looks like the function matches at x = 2. Do we have a jump discontinuity then?
Graphing it out, it doesn't look like we do. The function is the same number at the point where it switches, and we end up with a continuous whole function without any discontinuities. This is why we need to check for discontinuities even if it seems like it might have one.
What if the function was changed a little? Will it have a discontinuity then? Let's see.
In this case, we do have a discontinuity because where the first function ends at 3 at x = 2, the second function picks up at 4 at x = 2. There is a jump.
Remember to always check for a jump discontinuity even if it seems that a piecewise function has a discontinuity. A jump discontinuity looks as if the function literally jumped locations at certain values. There is no limit to the number of jump discontinuities you can have in a function. Functions that are broken up into separate regions are called piecewise functions. You can have as many regions as you want, as well.
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