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  • - [Instructor] What we're going to do in this video

  • is come up with a more rigorous definition for continuity.

  • And the general idea of continuity,

  • we've got an intuitive idea of the past,

  • is that a function is continuous at a point,

  • is if you can draw the graph of that function

  • at that point without picking up your pencil.

  • So what do we mean by that?

  • And this is...

  • What I just said is not that rigorous,

  • or not rigorous at all,

  • is that well, let's think about the point right over here.

  • Let's say that's RC.

  • If I can draw the graph at that point,

  • the value of the function at that point

  • without picking up my pencil, or my pen,

  • then it's continuous there.

  • So I could just start here,

  • and I don't have to pick up my pencil,

  • and there you go.

  • I can go through that point,

  • so we could say that our function is continuous there.

  • But if I had a function that looked

  • somewhat different that that,

  • if I had a function that looked like this,

  • let's say that it is defined up until then,

  • and then there's a bit of a jump,

  • and then it goes like this,

  • well this would be very hard to draw at...

  • This function would be very hard to draw

  • going through x equals c without picking up my pen.

  • Let's see, my pen is touching the screen,

  • touching the screen, touching the screen.

  • How do I keep drawing this function

  • without picking up my pen?

  • I would have to pick it up,

  • and then move back down here.

  • And so that is an intuitive sense

  • that we are not continuous in this case right over here.

  • Well let's actually come up

  • with a formal definition for continuity,

  • and then see if it feels intuitive for us.

  • So the formal definition of continuity, let's start here,

  • we'll start with continuity at a point.

  • So we could say the function f is continuous...

  • Continuous at x equals c.

  • If, and only if...

  • I'll draw this two-way arrow to show if, and only if,

  • the two-sided limit of f of x,

  • as x approaches c,

  • is equal to f of c.

  • So this seems very technical.

  • But let's just think about what it's saying.

  • It's saying look,

  • if the limit as we approach c

  • from the left and the right of f of x,

  • if that's actually the value of our function there,

  • then we are continuous at that point.

  • So let's look at three examples.

  • Let's look at one example, we are, we're...

  • By our picking-up-the-pencil idea,

  • it feels like we are continuous at a point.

  • And then let's think about a couple of examples

  • where it doesn't seem like we're continuous at a point,

  • and see how this more rigorous definition applies.

  • So, let's say that my function...

  • So let's say this right over here

  • is y is equal to f of x.

  • And, we care about the behavior right over here

  • when x is equal to c.

  • This is my X-axis, that's my Y-axis.

  • So we care about the behavior when x is equal to c.

  • And so, notice,

  • from our first intuitive sense,

  • I can definitely draw this function

  • as we go through x equals c without picking up my pencil,

  • so it feels continuous there.

  • There's no jumps or discontinuities that we can tell.

  • It just kind of keeps on going.

  • It seems all connected, is one way to think about it.

  • But let's think about this definition.

  • Well, the limit as x approaches c from the left, it is...

  • As we approach from the left,

  • it looks like it is approaching...

  • It looks like it is approaching f of c.

  • So this is the value, f of c right over here.

  • And as we approach from the right,

  • as we approach from the right,

  • it also looks like it's approaching f of c.

  • And we are defined right at x equals c.

  • And it is the value that we are approaching

  • from both the left or the right.

  • So this seems good in this scenario.

  • So now let's look at some scenarios

  • that we would have to pick up the pencil

  • as we draw the function through that point, through that...

  • Through that...

  • When x is equal to c.

  • So let's look at a scenario.

  • Let's look at a scenario where we have

  • what's often called a point discontinuity,

  • although you don't have to know at this point,

  • no pun intended, the formal terminology for it.

  • So let's say we have a function that...

  • Let's see, this is c.

  • Now let's say our function looks something...

  • Something like this.

  • So we go like this,

  • and at c let's say it's equal to that.

  • So, f of c is right over here.

  • F of c would be that value.

  • But what's the limit as x approaches c?

  • So the limit as x approaches c,

  • this would be a two-sided limit of f of x.

  • Well, this is, as we approach from the left,

  • it looks like we are approaching this value right over here.

  • And from the right,

  • it looks like we are approaching that same value.

  • And so, we could call that L.

  • And L is different than f of c.

  • And so, in this case, by our formal definition,

  • we will not be continuous at, for...

  • F will not be continuous for x is equal...

  • Or at the point x...

  • Or when x is equal to c.

  • And you can see that there.

  • If we try to draw this, okay,

  • my pencil is touching the paper,

  • touching the paper, touching the paper.

  • Uh oh, if I needed to keep drawing this function,

  • I'd have to pick up my pencil, move it over here,

  • then pick it up again and then jump right back down.

  • And but this rigorous definition

  • is giving us the same conclusion.

  • The limit as we approach x equals c

  • from the left and the right,

  • it's a different value than f of c.

  • And so, this is not continuous.

  • Not...

  • Not continuous.

  • And let's think about another scenario.

  • Let's think about a scenario...

  • And actually, maybe let's think about a scenario

  • where the limit...

  • The two-sided limit doesn't even exist.

  • So, there are my axes, x and y.

  • And let's say it's doing something like this.

  • Let's say it's doing something like this,

  • and that it does something like this and goes like that.

  • And let's say that this right over here is our c.

  • And so let's see, this is f of c right over here.

  • That is...

  • Lemme draw a little bit neater.

  • That is f of c.

  • And it does look like the limit,

  • as x approaches c from the left,

  • so from values less than c,

  • it does look like that is approaching f of c.

  • But if we look at the limit

  • as x approaches c from the right,

  • that looks like it's approaching some other value.

  • That looks like it's approaching this value right over here,

  • let's call it L.

  • That's approaching L,

  • and L does not equal f of c.

  • And so in this situation,

  • the two-sided limit doesn't even exist.

  • We're approaching two different values

  • when we approach from the left and from the right.

  • And since so the limit doesn't even exist at c,

  • this is definitely not going to be continuous.

  • And this matches up to our expectations

  • with our little do-we-have-to-pick-up-the-pencil test.

  • If I have to draw this, I can leave my pencil,

  • it's on the paper, it's on the paper,

  • it's on the paper, it's on the paper.

  • How am I going to continue to draw this function,

  • this graph of the function, without picking up my pencil?

  • Pick it up, put it back down, and then keep drawing it.

  • And so once again, this right over here is not continuous.

  • Both intuitively, by our pick-up-the-pencil definition,

  • and also by this more rigorous definition where,

  • in this case the limit,

  • the two-sided limit at x equals c doesn't even exist,

  • so we're definitely not gonna be continuous.

  • But even when the two-sided limit does exist,

  • but the limit is a different value

  • than the value of the function,

  • that will also not be continuous.

  • The only situation that it's going to be continuous

  • is if the two-sided limit approaches the same value

  • as the value of the function.

  • And if that's true, then we're continuous.

  • If we're continuous, that is going to be true.

- [Instructor] What we're going to do in this video

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Continuity at a point | Limits and continuity | AP Calculus AB | Khan Academy

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    yukang920108 に公開 2022 年 07 月 05 日
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