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AP®︎ Calculus AB (2017 edition)
Course: AP®︎ Calculus AB (2017 edition) > Unit 3
Lesson 13: Optional videos- Justifying the power rule
- Proof of power rule for positive integer powers
- Proof of power rule for square root function
- Limit of sin(x)/x as x approaches 0
- Limit of (1-cos(x))/x as x approaches 0
- Proof of the derivative of sin(x)
- Proof of the derivative of cos(x)
- Product rule proof
- Proof: Differentiability implies continuity
- If function u is continuous at x, then Δu→0 as Δx→0
- Chain rule proof
- Quotient rule from product & chain rules
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Proof: Differentiability implies continuity
Sal shows that if a function is differentiable at a point, it is also continuous at that point.
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Does the opposite also imply ie. continiuity implies differentiability ? if so why and why not ?(40 votes)
No, continuity does not imply differentiability. For instance, the functionƒ: R → Rdefined byƒ(x) = |x|is continuous at the point0, but it is not differentiable at the point0.
It can get worse. See for instance:
http://en.wikipedia.org/wiki/Weierstrass_function
http://mathworld.wolfram.com/WeierstrassFunction.html(50 votes)
When Sal starts his proof of how differentiability implies continuity, why does he only use the numerator (f(x)-f(c)) and drops the denominator (x-c)??(30 votes)
This is a common way of making demonstrations in mathematics: you start from an expression A, and by simplifying it you show that it is equal to an expression B. Then starting from A again, you show that it also simplifies to an expression C. You can then say that B = C. The whole point is that, by starting from A, you only make simplifications, but if you had to start from another expression, you would have to go through more complicated steps to reach A.(17 votes)
Everything is very clear except for the part where we "decide to take a look at a slightly different limit " which is lim { f(x) - f(c) }. How do we decide to take a look at it... it seems like it is "planned" (so that later we just multiply it and divide it by (x-c). If you understand what I am trying to say. Or is it assumed at that point that the lim { f(x) - f(c) } is the same as if we wrote lim f(x) = f(c) which is the definition of continuity ?(15 votes)- We indeed derive lim x→c { f(x) - f(c) } from lim x→c f(x) = f(c). Because f(c) is a constant, after removing it from both sides we can put it inside the limit.
lim x→c f(x) = f(c)
[lim x→c f(x)] - f(c) = 0
lim x→c {f(x) - f(c)} = 0
Edit (see comment): This is not a proof but an explanation for the derivation of the expression which will in turn proove that differentiablity implies continuity.(3 votes)
Why did he use limit x>c f(x)-f(c)?(9 votes)- At3:15Sal writes defines continuity as lim x→c f(x) = f(c).
He then uses lim x→c f(x)-f(c) and shows this equals zero. Let's see what this gives us:
lim x→c f(x) - f(c) = 0
[lim x→c f(x)] - [lim x→c f(c)] = 0
lim x→c f(x) = lim x→c f(c)
Now, the right-hand side is just f(c) because it doesn't have x in it. So we've got:
lim x→c f(x) = f(c)
which is our definition of continuity, which we wanted to show.(8 votes)
- When Sal says a limit "equals" something, that is a simplification or a less formal way of stating what is happening correct? If I am not mistaken, I thought limits only "tend toward" or "go to" a value, but never actually reaching that value all the way.(5 votes)
The limit is a fixed value. Like 1. A stationary value or object cannot approach anything, but it can be approached. X is the value that is changing. We plug something in, and it will work along that function AS LONG AS IT IS NOT THE LIMIT. It can get very close, of course.
Think of the way we say the notation, "What is the LIMIT of f(x) as X approaches 'some value?'".(13 votes)
I understand the proof but what about this example? Let's say a car is moving along the road, time is independent variable and distance is dependent. Then velocity at some moment of time has to be the derivative. At some moment car hits a wall, crashes into it and loses all the velocity (completely ineslastic collision). At the very last moment the car had some velocity (differentiable) but suddenly lost it afterwards (not continuous). Where's the mistake?(5 votes)
On a small enough scale, the velocity of each particle of the car will experience an electromagnetic force with the particles in the wall. Provided that the wall is sturdy enough to handle the collision, each individual particle will feel an abrupt acceleration forcing them to slow down and recoil. This is all one continuous motion, but it takes place on a small scale of both position and time.(7 votes)
I wonder in the first graph where f(c) is a point other than on the curve, here f(x) is defferenciable at c, that means limf(x) - f(c) =0, which is saying f(x) is continuous at c, obviously it's not right, how can explain that?(4 votes)
If f(c) is not on the curve of the function then that is a discontinuity and the function is not differentiable at that point.(3 votes)
- 10:09: Is assumption correct ?
We assumed thatlim (x > c) f(c) = f(c)
We can only assume this when we already know that f is continuous at c.
But the whole point of this proof was to show differentiability implies continuity.
Am I missing something here ?(4 votes)
In order for 𝑓(𝑥) to be differentiable at 𝑥 = 𝑐 the function must first of all be defined for 𝑥 = 𝑐, and since differentiability is a prerequisite for the proof we thereby know that 𝑓(𝑐) is indeed a constant, and so
lim(𝑥 → 𝑐) 𝑓(𝑐) = 𝑓(𝑐)(3 votes)
What if you had a jump in discontinuity, but the slope remained the same, the function was just shifted up or down at the point? Then if you took the limit from the positive side and from the negative, they would give you the same value.(4 votes)
They wouldn't. The limit is the number that the function's output approaches, not the slope that the curve has. So when you shift up or down at the point (say x = c), you're jumping to a completely different number, which means the number is different when approaching from either side of c. So the limit doesn't exist.(2 votes)
Multiplying with (x-c)/(x-c) where c approaches x... isn't that multiplying by 0/0? How is this correct? Of course this is the limit and limit never really is c but it gets close to it, so how can we say it's zero? Unless of course we actually substitute the value of that limit which is c? How do I know when to substitute c into x? This is rather confusing for me. So basically, a limit never really gets to c but it's equal to a constant number which is that limit if we were to put c into x? This constant is basically what the limit evaluates to in, conclusion? Also at the very beginning, when he needed to evalute limit of f(x)-f(c) as x approaches c, couldn't he evaluated it at the very beginning to be equal to zero? Or is it that at this point in time, we do not know whether the function is continous or not? So this is why we multiplied by (x-c), as this can be more easily evaluated, it does not change our original expression and it also gives a new value which is d/dx of f?(2 votes)
Remember that while x approaches c, it never actually gets there (and in fact we don't really care about what happens at x = c). Thus we are never actually dividing by 0, only by an infinitesimally small number close to 0.(5 votes)
3:15Video transcript
- [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.