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

  • is prove that if a function is differentiable

  • at some point, C, that it's also

  • going to be continuous at that point C.

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

  • what differentiability means and what continuity means.

  • So, first, differentiability.

  • Differentiability

  • So, let's think about that, first.

  • And it's always helpful to draw ourselves

  • a function.

  • So, that's our Y-axis.

  • This is our X-axis.

  • And let's just draw some function, here.

  • So, let's say my function looks like this

  • and we care about the point X equals C,

  • which is right over here.

  • So, that's the point X equals C,

  • and then, this value, of course,

  • is going to be F of C.

  • F of C.

  • 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.

  • Say, some arbitrary X out here.

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

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

  • this Y value, would be F of X.

  • Would be F of X.

  • 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.

  • And as X approaches C, this secant,

  • the slope of the secant line is going to approach

  • the slope of the tangent line, or,

  • it's going to be the derivative.

  • And so, we could take the limit...

  • The limit as X approaches C,

  • as X approaches C,

  • of the slope of this secant line.

  • So, what's the slope?

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

  • The change in Y is F of X minus F of C,

  • that's our change in Y right over here.

  • 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.

  • It is X minus,

  • 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,

  • F prime of C.

  • All of this is review.

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

  • if we're saying that the function, F,

  • is differentiable at X equals C,

  • we're really just saying that this limit

  • right over here actually exists.

  • And if this limit actually exists,

  • we just call that value F prime of C.

  • So, that's just a review of differentiability.

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

  • Continuity.

  • So, the definition for continuity is

  • if the limit as X approaches C of F of X

  • is equal to F of C.

  • 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,

  • well, let's visualize it and hopefully

  • it'll make some intuitive sense.

  • So, if you have a function,

  • so, let's actually look at some cases

  • where you're not continuous.

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

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

  • so, this is X equals C, so, if you had

  • a point discontinuity, so,

  • lemme draw it like this, actually.

  • So, you have a gap, here, and X equals,

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

  • So, this is F of C, and then,

  • the function continues like this.

  • The limit, as X approaches C of F of X

  • is going to be this value, which is clearly different

  • than F of C.

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

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

  • you're approaching this value.

  • This, right over here, is the limit,

  • 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,

  • you have a point discontinuity.

  • So, for at least in this case, our,

  • this definition of continuity would properly

  • identify this as not a continuous function.

  • Now, you could also think about a jump discontinuity.

  • You can also think about a jump discontinuity.

  • So, let's look at this.

  • And all this is, hopefully, a little bit of review.

  • So, a jump discontinuity at C, at X equals C,

  • might look like this.

  • Might look like this.

  • So, this is at X equals C.

  • So, this is X equals C right over here.

  • This would be F of C.

  • But, if you tried to find a value at the limit

  • as X approaches C of F of X,

  • you'd get a different value as you approach

  • C from the negative side, you would approach this value,

  • and as you approach C from the positive side,

  • you would approach F of C, and so, the limit wouldn't exist.

  • So, this limit right over here wouldn't exist

  • in the case of jump, of this type of a jump discontinuity.

  • So, once again, this definition would properly

  • say that this is not, this one right over here,

  • is not continuous, this limit actually would not even exist.

  • And then, you could even look at a,

  • you could look at a function that is truly continuous.

  • If you look at a function that is truly continuous.

  • So, something like this.

  • Something like this.

  • That is X equals C.

  • Well, this is F of C.

  • This is F of C.

  • And if you were to take the limit as X approaches C,

  • as X approaches C from either side of F of X,

  • you're going to approach F of C.

  • So, here, you have the limit as X approaches C

  • of F of X, indeed, is equal to F of C.

  • So, it's what you would expect for a continuous function.

  • So, now that we've done that review

  • of differentiability and continuity,

  • let's prove that differentiability

  • actually implies continuity, and I think it's important

  • to kinda do this review, just so that you can

  • really visualize things.

  • So, differentiability implies this limit

  • right over here exists.

  • So, let's start with a slightly different limit.

  • Lemme draw a line, here, actually.

  • Lemme draw a line just so we're doing something different.

  • So, let's take, let us take the limit

  • as X approaches C of F of X,

  • of F of X minus F of C.

  • Of F of X minus F of C.

  • Well, can we rewrite this?

  • Well, we could rewrite this as the limit,

  • as X approaches C, and we could essentially

  • take this expression and multiply

  • and divide it by X minus C.

  • So, let's multiply it times X minus C.

  • X minus C, and divide it by X minus C.

  • So, we have F of X minus F of C,

  • all of that over X minus C.

  • So, all I did is I multiplied and I divided

  • by X minus C.

  • Well, what's this limit going to be equal to?

  • This is going to be equal to,

  • it's going to be the limit, and I'm just applying

  • the property of limit, applying a property of limits, here.

  • So, the limit of the product is equal to

  • the same thing as a product of the limits.

  • So, it's the limit as X approaches C of X minus C,

  • times the limit, lemme write this way,

  • times the limit as X approaches C

  • of F of X minus F of C,

  • all of that over X minus C.

  • Now, what is this thing right over here?

  • Well, if we assume that F is differentiable at C,

  • and we're going to do that, actually,

  • I should have started off there.

  • Let's assume 'cause we wanted to show the differentiability,

  • it proves continuity.

  • If we assume F differentiable,

  • differentiable at C, well then,

  • this right over here is just going to be F prime of C.

  • This right over here, we just saw it right over here,

  • that's this exact same thing.

  • This is F prime, F prime of C.

  • And what is this thing right over here?

  • The limit as X approaches C of X minus C?

  • Well, that's just gonna be zero.

  • As X approaches C, there's gonna become,

  • approach C minus C, that's just going to be zero.

  • So, what's zero times F prime of C?

  • Well, F prime of C is just going to be some value,

  • so, zero times anything is just going to be zero.

  • So, I did all that work to get a zero.

  • Now, why is this interesting?

  • Well, we just said, we just assumed

  • that if F is differentiable at C,

  • and we evaluate this limit, we get zero.

  • So, if we assume F is differentiable at C,

  • we can write, we can write the limit,

  • I'm just rewriting it, the limit as X approaches C

  • of F of X minus F of C, and I could even put

  • parenthesis around it like that,

  • which I already did up here,

  • is equal to zero.

  • Well, this is the same thing,

  • I could use limit properties again,

  • this is the same thing as saying,

  • and I'll do it over, well, actually,

  • lemme do it down here.

  • The limit as X approaches C of F of X

  • minus the limit as X approaches C

  • of F of C, of F of C, is equal to zero.

  • The different, the limit of the difference

  • is the same thing as the difference of the limits.

  • Well, what's this thing over here going to be?

  • Well, F of C is just a number,

  • it's not a function of X anymore,

  • it's just, F of C is going to valuate it to something.

  • So, this is just going to be F of C.

  • This is just going to be F of C.

  • So, the limit of F of X as X approaches C,

  • minus F of C is equal to zero.

  • Well, just add F of C to both sides and what do you get?

  • Well, you get the limit as X approaches C

  • of F of X is equal to F of C.

  • And this is the definition of continuity.

  • The limit of my function as X approaches C

  • is equal to the function, is equal to the value

  • of the function at C.

  • This is, this means that our function is continuous.

  • Continuous at C.

  • So, just a reminder, we started assuming

  • F differentiable at C, we use that fact

  • to evaluate this limit right over here,

  • which, we got to be equal to zero,

  • and if that limit is equal to zero,

  • then, it just follows, just doing a little bit of algebra

  • and using properties of limits,

  • that the limit as X approaches C of F of X

  • is equal to F of C, and that's our definition

  • of being continuous.

  • Continuous at the point C.

  • So, hopefully, that satisfies you.

  • If we know that the derivative exists at a point,

  • if it's differentiable at a point C,

  • that means it's also continuous at that point C.

  • The function is also continuous at that point.

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

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

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    yukang920108 に公開 2022 年 07 月 12 日
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