Name: Proof: Differentiability implies continuity | Derivative rules | AP Calculus AB | Khan Academy
Uploaded: 2022-07-12T12:46:41.000Z
Duration: 11 min 39 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...

To-infinitive

- [Voiceover] What I hope to do in this video

is prove that if a function is differentiable

But, before we do the proof, let's just remind ourselves

what differentiability means and what continuity means.

And it's always helpful to draw ourselves

So, let's say my function looks like this

And one way that we can find the derivative

at X equals C, or the slope of the tangent line

at X equals C is, we could start with some other point.

So, let's say this is some arbitrary X out here.

So, then, this point right over there, this value,

This graph, of course, is a graph of Y equals F of X.

And we can think about finding the slope of this line,

this secant line between these two points,

but then, we can find the limit as X approaches C.

the slope of the secant line is going to approach

Well, it's gonna be change in Y over change in X.

This is all review, this is just one definition

of the derivative, or one way to think about the derivative.

So, it's going to be F of X minus F of C,

that's our change in Y, over our change in X.

Over our change in X, which is X minus C.

So, if this limit exists, then, we're able to find

the slope of the tangent line at this point,

and we call that slope of the tangent line,

we call that the derivative at X equals C.

We say that this is going to be equal to F prime,

So, if we're saying, one way to think about it,

So, that's just a review of differentiability.

Now, let's give ourselves a review of continuity.

Now, this might seem a little bit, you know,

well, it might pop out to you as being intuitive

or it might seem a little, well, where did this come from,

And that actually might make it a little bit more clear.

So, if you had a point discontinuity at X equals C,

when X equals C, F of C is actually way up here.

is going to be this value, which is clearly different

This value right over here, if you take the limit,

if you take the limit as X approaches C of F of X,

as X approaches C of F of X, which is different than F of C.

So, it makes it, so, this definition of continuity

seems to be good, at least for this case,

because this is not a continuous function,

Subtitles ListPlay Video

Proof: Differentiability implies continuity | Derivative rules | AP Calculus AB | Khan Academy

essentially

assume

approach

properly