In this lesson, we talk about the types of discontinuities that you commonly see in functions. In particular, learn how to identify point, jump and asymptotic discontinuities.
Let's talk about types of discontinuities by considering UFO behavior. Remember that a discontinuity is where the value of a function jumps, and a function is continuous if you can trace the entire graph without lifting your finger from the paper.
Consider a UFO who's hanging around in Earth - maybe he's checking out crops. At some point in time, it appears that he jumps to the Moon - but just for a single instant. Maybe a little later in time, he jumps up to Mars but, again, only for that exact moment in time. What we've done is we've put holes in the graph where he's not at a particular altitude at a particular time - but only at that instant. And we put a point on the graph for where he is at that particular instant.
Graph showing a point discontinuity
Each point is a single point in time. I keep saying the word point, so I should call these point discontinuities or, if you want to get a little more formal, removable discontinuities.
Now, for example, what about f(x) where f(x)=1 when x<1 or x>1, and f(x)=2 for x=1. So at that single point, x=1, our function has the value 2, but everywhere else, the function has the value 1. So this is a point, or removable, discontinuity at x=1.
Let's consider the same UFO now, but instead of checking out the Moon at a single point in time, he jumps up to the Moon. So he disappears from our crops and reappears up at the Moon and then he stays there. At some point in time, his location jumps. Understandably, we call these jump discontinuities. For jump discontinuities, the function jumps to a new value.
Let's look at a mathematical example. Let's look at a function where f(x)=1 for x less than or equal to zero, and when x>0, all of a sudden, f(x)=x. We're looking at the right side of the graph now, and at the point x=0, we have a jump in our function.
Graph showing a jump discontinuity
Another example is the function of the floor of x, f(x)=floor(x). So floor(x) rounds down, such that the floor of 4.1 is 4 and the floor of 4.9 is 4 and floor of 4.999999 is 4. You can also think of this as cutting off everything to the right-hand side of the decimal point. If I graph floor(x), then I get a stair-step pattern. If x<5 but x is greater than or equal to 4, the value is 4. If x=2.3, then f(x)=2. Now, at every integer, the function jumps, so you can think of it as having an infinite number of jump discontinuities.
Okay, we have point discontinuities and jump continuities, so what happens if the UFO takes a nosedive and actually goes into the Earth, maybe toward the center of the Earth. And, all of sudden you see him way up out in space and flying back down to check out crops again. Maybe there's some kind of wormhole. At that moment in time, we have a vertical asymptote. So this type of discontinuity, where we're lifting up our finger and putting it down somewhere else, is called an asymptotic discontinuity.
Let's look at a mathematical example. Say f(x)=1/x, and we know that the graph looks like this. At x=0, we have a vertical asymptote, and we have an asymptotic discontinuity. What about the function f(x)=x^-2? Well, this is really saying that f(x)=1/(x^2). We have the same problem; this is not defined at x=0.
Graph showing an asymptotic discontinuity
Let's look at something a little more complex. This graph looks very busy. I don't know what this function is, but let's just analyze the graph. At x=1, there's a point discontinuity, or removable discontinuity. At x=2, we have a jump discontinuity, because the function jumps at that particular value of x. At x=3, there is no discontinuity. Sure, there's a kind of corner in the graph, but I can trace this line without lifting my finger up, so this is continuous. And, at x=5, we have an asymptotic discontinuity, where the function approaches minus infinity and then reappears at infinity. So we have all three kinds of discontinuities: points, jumps and asymptotic.
To recap, there are three types of discontinuities for f(x).
The point, or removable, discontinuity is only for a single value of x, and it looks like single points that are separated from the rest of a function on a graph. A jump discontinuity is where the value of f(x) jumps at a particular point. An asymptotic discontinuity is where the value of f(x) goes to +/- infinity at a particular point in x.