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

  • is explore the notion of differentiability at a point.

  • And that is just a fancy way of saying

  • does the function have a defined derivative at a point?

  • So let's just remind ourselves a definition of a derivative.

  • And there's multiple ways of writing this.

  • For the sake of this video, I'll write it as

  • the derivative of our function at point C,

  • this is Lagrange notation with this F prime.

  • The derivative of our function F at C is going to be equal

  • to the limit as X approaches Z

  • of F of X, minus F of C,

  • over X minus C.

  • And at first when you see this formula,

  • and we've seen it before, it looks a little bit strange,

  • but all it is is it's calculating the slope,

  • this is our change in the value of our function,

  • or you could think of it as our change in Y,

  • if Y is equal to F of X, and this is our change in X.

  • And we're just trying to see, well, what is that slope

  • as X gets closer and closer to C, as our change in X

  • gets closer and closer to zero?

  • And we talk about that in other videos.

  • So I'm now going to make a few claims in this video,

  • and I'm not going to prove them rigorously.

  • There's another video that will go a little bit more

  • into the proof direction.

  • But this is more to get an intuition.

  • And so the first claim that I'm going to make

  • is if F is differentiable,

  • at X equals C,

  • at X equals C,

  • then F is continuous

  • at X equals C.

  • So I'm saying if we know it's differentiable,

  • if we can find this limit,

  • if we can find this derivative at X equals C,

  • then our function is also continuous at X equals C.

  • It doesn't necessarily mean the other way around,

  • and actually we'll look at a case where it's not

  • necessarily the case the other way around that

  • if you're continuous, then you're definitely differentiable.

  • But another way to interpret what I just wrote down is,

  • if you are not continuous, then you definitely

  • will not be differentiable.

  • If F not continuous

  • at X equals C, then F

  • is not differentiable,

  • differentiable at X is equal to C.

  • So let me give a few examples of a non-continuous function

  • and then think about would we be able to find this limit.

  • So the first is where you have a discontinuity.

  • Our function is defined at C, it's equal to this value,

  • but you can see as X becomes larger than C,

  • it just jumps down and shifts right over here.

  • So what would happen if you were trying to find this limit?

  • Well, remember, all this is is a slope of a line

  • between when X is some arbitrary value,

  • let's say it's out here, so that would be X,

  • this would be the point X comma F of X,

  • and then this is the point C comma F of C right over here.

  • So this is C comma F of C.

  • So if you find the left side of the limit right over here,

  • you're essentially saying okay, let's find this slope.

  • And then let me get a little bit closer,

  • and let's get X a little bit closer

  • and then let's find this slope.

  • And then let's get X even closer than that

  • and find this slope.

  • And in all of those cases, it would be zero.

  • The slope is zero.

  • So one way to think about it, the derivative or this limit

  • as we approach from the left, seems to be approaching zero.

  • But what about if we were to take Xs to the right?

  • So instead of our Xs being there,

  • what if we were to take Xs right over here?

  • Well, for this point, X comma F of X,

  • our slope, if we take F of X minus F of C

  • over X minus C, that would be the slope of this line.

  • If we get X to be even closer, let's say right over here,

  • then this would be the slope of this line.

  • If we get even closer, then this expression

  • would be the slope of this line.

  • And so as we get closer and closer to X being equal to C,

  • we see that our slope is actually

  • approaching negative infinity.

  • And most importantly, it's approaching

  • a very different value from the right.

  • This expression is approaching a very different value

  • from the right as it is from the left.

  • And so in this case, this limit up here won't exist.

  • So we can clearly say this is not differentiable.

  • So once again, not a proof here.

  • I'm just getting an intuition for if something

  • isn't continuous, it's pretty clear, at least in this case,

  • that it's not going to be differentiable.

  • Let's look at another case.

  • Let's look at a case where we have what's sometimes called

  • a removable discontinuity or a point discontinuity.

  • So once again, let's say we're approaching from the left.

  • This is X, this is the point X comma F of X.

  • Now what's interesting is where as this expression

  • is the slope of the line connecting X comma F of X

  • and C comma F of C, which is this point, not that point,

  • remember we have this removable

  • discontinuity right over here,

  • and so this would be this expression is calculating

  • the slope of that line.

  • And then if X gets even closer to C, well, then we're gonna

  • be calculating the slope of that line.

  • If X gets even closer to C, we're gonna be calculating

  • the slope of that line.

  • And so as we approach from the left, as X approaches C

  • from the left, we actually have a situation

  • where this expression right over here is going

  • to approach negative infinity.

  • And if we approach from the right, if we approach

  • with Xs larger than C, well, this is our X comma F of X,

  • so we have a positive slope and then as we get closer,

  • it gets more positive, more positive

  • approaches positive infinity.

  • But either way, it's not approaching a finite value.

  • And one side is approaching positive infinity,

  • and the other side is approaching negative infinity.

  • This, the limit of this expression, is not going to exist.

  • So once again, I'm not doing a rigorous proof here,

  • but try to construct a discontinuous function

  • where you will be able to find this.

  • It is very, very hard.

  • And you might say, well, what about the situations

  • where F is not even defined at C, which for sure

  • you're not gonna be continuous if F is not defined at C.

  • Well if F is not defined at C, then this part

  • of the expression wouldn't even make sense,

  • so you definitely wouldn't be differentiable.

  • But now let's ask another thing.

  • I've just given you good arguments for when you're

  • not continuous, you're not going to be differentiable,

  • but can we make another claim that if you are continuous,

  • then you definitely will be differentiable?

  • Well, it turns out that there are for sure

  • many functions, an infinite number of functions,

  • that can be continuous at C, but not differentiable.

  • So for example, this could be an absolute value function.

  • It doesn't have to be an absolute value function,

  • but this could be Y is equal to

  • the absolute value of X minus C.

  • And why is this one not differentiable at C?

  • Well, think about what's happening.

  • Think about this expression.

  • Remember, this expression all it's doing is calculating

  • the slope between the point X comma F of X

  • and the point C comma F of C.

  • So if X is, say, out here, this is X comma F of X,

  • it's going to be calculated, so if we take the limit

  • as X approaches C from the left,

  • we'll be looking at this slope.

  • And as we get closer, we'll be looking at this slope

  • which is actually going to be the same.

  • In this case it would be a negative one.

  • So as X approaches C from the left,

  • this expression would be negative one.

  • But as X approaches C from the right,

  • this expression is going to be one.

  • The slope of the line that connects these points is one.

  • The slope of the line that connects these points is one.

  • So the limit of this expression, or I would say the value

  • of this expression, is approaching two different values

  • as X approaches C from the left or the right.

  • From the left, it's approaching negative one,

  • or it's constantly negative one and so it's approaching

  • negative one, you could say.

  • And from the right, it's one,

  • and it's approaching one the entire time.

  • And so we know if you're approaching two different values

  • from on the left side or the right side of the limit,

  • then this limit will not exist.

  • So here, this is not, not differentiable.

  • And even intuitively, we think of the derivative

  • as the slope of the tangent line.

  • And you could actually draw an infinite number

  • of tangent lines here.

  • That's one way to think about it.

  • You could say, well, maybe this is the tangent line

  • right over there, but then why can't I make

  • something like this the tangent line?

  • That only intersects at the point C comma zero.

  • And then you could keep doing things like that.

  • Why can't that be the tangent line?

  • And you could go on and on and on.

  • So the big takeaways here, at least intuitively,

  • in a future video I'm going to prove to you

  • that if F is differentiable at C that it is continuous at C,

  • which can also be interpreted as that if you're not

  • continuous at C, then you're not gonna be differentiable.

  • These two examples will hopefully give you

  • some intuition for that.

  • But it's not the case that if something is continuous

  • that it has to be differentiable.

  • It oftentimes will be differentiable, but it doesn't have

  • to be differentiable, and this absolute value function

  • is an example of a continuous function at C,

  • but it is not differentiable at C.

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

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Differentiability and continuity | Derivatives introduction | AP Calculus AB | Khan Academy

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    yukang920108 に公開 2022 年 07 月 12 日
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