Erin has been writing and editing for several years and has a master's degree in fiction writing.
Can Earth ever compete with extraterrestrial UFOs? In this lesson, you'll learn that not all functions are continuous, but most have regions where they are continuous. Discover how to define regions of continuity for functions that have discontinuities.
Regions of Continuity
The jets can follow the UFOs along regions of continuity
Let's take another look at UFOs and Earth's defense jets. UFOs can travel on paths that are not continuous. That is, they can jump from one place to another without traveling through the places in between. If we graph the altitude as a function of time, then the jumps are located at discontinuities. They are followed by our Earth defense jets, which can't make such jumps. These jets, follow a continuous path.
Hang on a second. If the jets are following continuous paths, or continuous functions, and the UFOS have discontinuous paths where they've jumped from one altitude to another, can the jets ever hope to trail the UFOs? Well, sure. The jets can follow the UFOs between their discontinuities. They can follow the UFOs when the UFOs paths are continuous. We call these paths regions of continuity. A region of continuity is where you have a function that is continuous. It's only that region in x that f(x) is continuous. That is, when you can trace f(x) without lifting your finger.
Regions around Jumps
f(x) has regions of continuity for x is less than and greater than 1
Let's take a look at an example. Say you have some function: y=f(x). That function equals 1 when x < 1, it equals 2 when x=1 and for all values of x greater than 1, the function f(x)=3. So this is as if you have your UFO and all of a sudden at time t=1, it jumps up to the Moon and then to Mars, instantaneously. Then he just hovers out at Mars for a while. In this case, we say that f(x) has two regions of continuity. One for x < 1 and one for x > 1. Because if I'm tracing the line f(x) for all values less than 1, I don't have to lift my finger up from this graph. The same thing after 1. For any x greater than 1, I can trace this line without lifting my finger from the graph.
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Let's take a look at another example. Say you have the function f(x)=x for x < 4, and 8 - x for x > or equal to 4. Now I can graph this out, and I can trace the entire line without lifting my finger from the paper, even at the point that we call a kink. In this case, since I can trace the entire function without lifting my finger from the graph, this entire function is continuous. The region of continuity is all of x.
The arrow on the graph shows the location of the kink for this continuous function
Considering Multi-Step Functions
Let's look at another example. Say you're given f(x) equals:
x when x < 2
1 when x=2
x when 2 < x < 3
3 when x > or equal to 3
I don't know where the regions of continuity are, so I'm going to graph this out. First, I'm going to graph x for all values of x less than 2. At x=2, f(x)=1. That's a jump, so I know that I have a region of continuity for x < 2, but not at x=2. From 2 to 3, my function f(x) equals x again. At 3 and values greater than 3, my function f(x)=3. I can trace the entire function without lifting my finger from the paper. So for all values greater than 2, my function is also continuous. I've got two regions of continuity here: x < 2 and x > 2. At x=2, there's this point or removable discontinuity.
Graph for the final multi-step function example
Let's bring this back to our UFO and Earth's defense system, the fighter jets. If we take a look at the altitudes of the UFOs as a function of time, there are certain regions where the function (their height) is continuous. That is, I can trace it with my finger. In these regions where the function is continuous, our fighter jets can actually follow them. We call these regions of continuity. To recap, regions of continuity are the values of x where some function, f(x), is continuous.
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