Name: Differentiability and continuity | Derivatives introduction | AP Calculus AB | Khan Academy
Uploaded: 2022-07-12T12:22:49.000Z
Duration: 9 min 38 s
Description: Thousands of YouTube videos with English-Chinese subtitles! Now you can learn to understand native speakers, expand your vocabulary, and improve your pronunciation...

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

is explore the notion of differentiability at a point.

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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

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

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

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

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

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

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

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

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

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's get X even closer than that

And in all of those cases, it would be 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?

what if we were to take Xs right over here?

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

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

This expression is approaching a very different value

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

So we can clearly say this is not differentiable.

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 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.

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

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