Travel to space and explore the difference between continuous and discontinuous functions in this lesson. Learn how determining continuity is as easy as tracing a line.
Continuity: UFOs and Jets
jpg” alt=”Polynomial Continuous Function” />
Let’s talk for a minute about continuity. Hey, what’s that? It looks like a UFO! Have you ever noticed that UFOs can be hanging around, checking out Earth for a minute, and then all of a sudden the next instant they’re at the Moon? I mean, what’s going on there? If I take a look at their altitude as a function of time, they can be here checking out crops, and then BOOM – up on the Moon, or perhaps fighting another UFO way up in the upper atmosphere. And then just for an instant, they leave the Moon, come back down here and say, ‘Hey, are those crops still here?’ And go back up to the Moon that very instant. They’re just jumping all over the place.Now let’s say that we have some jets – the Earth Defense Jets – whose sole job is to track these UFOs and to find out where they’re going. Let’s say when the UFOs are down here, checking out our crops and checking out Earth, our Earth Defense Jets take off and they get up to the altitude of the UFOs.
Then all of a sudden when the UFOs hop up to the Moon, the Earth Defense Jets go, ‘Well, where did you guys go? Oh, they’re way up at the Moon!’ So they slowly fly up to the Moon (these are amazing jets here), and then in that instant when the UFO pops back down to Earth, the Earth Defense Jets sit there and go, ‘Where did they go? Ah, they’re down there.’ But in that same instant right after the UFOs pop down and see that the crops are there, they pop back up to the Moon and the Earth Defense Jets go, ‘Okay, we’re good, we can track you up here.’Why is it that the Earth Defense Jets can’t follow the UFOs?
Continuous and Discontinuous Functions
Well they can’t follow the UFOs exactly because the UFOs follow a discontinuous path. One instant they’re down on Earth checking out the crops, and then BOOM, they’re up at the Moon. They don’t really have to travel between the two, they’re just up there. The Earth Defense Jets don’t have that ability; they can’t just disappear one place and appear somewhere else.
They follow a continuous path. They can’t pop back down to follow the UFOs.Let’s make this a little bit more formal and say that continuous functions – like the Earth Defense Jets – do not jump around.
If you graph one, you can actually trace the entire function with your finger without lifting it up from the paper. So in the case of a UFO, you couldn’t trace its entire path without lifting your finger. When it hopped up to the Moon, if you were to trace its path, you’d have to lift your finger from the paper and put it back down on the Moon.Let’s look at a few examples.
Let’s look at f(x)=1/(x – 1). Now If I graph this out, I actually have an asymptote at 1. So as x approaches 1 from 0 lets say, f(x) goes to negative infinity. Well at some point, I’m going to have to lift my finger up, put it back down at positive infinity once I get to the other side of 1 (just greater than 1), and I can trace the rest of the function.
So again, this is discontinuous because I have to lift my finger up from the paper.
Let’s review. Continuity. Now a function is continuous if you can trace the entire function on a graph without picking up your finger. This is kind of like Earth’s Defense Jets; if you trace their altitude as a function of time, you never have to lift your finger up from the sheet of paper. If on the other hand, you traced the UFO’s altitude as a function of time, you might have to lift your finger up at various points in time, when the UFO is hopping around from our crops to the Moon, from our crops to Mars; when they do that instantaneous hop, that’s a discontinuity and you have to lift your finger up from the paper.