Discontinuities in Functions and Graphs Video

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  • 0:06 Discontinuity
  • 1:01 Point Discontinuity
  • 2:00 Jump Discontinuity
  • 3:41 Asymptotic Discontinuity
  • 5:54 Lesson Summary
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Lesson Transcript
Instructor: Robert Egan
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.

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
Graph showing Point Discontinuity

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.

Jump Discontinuity

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
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.

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