Differentiability and continuity (video) | Khan Academy
Why is it that all differentiable functions are continuous but not all continuous functions are differentiable? A continuous function is a function whose graph is a single unbroken curve. What does this tell us about the relationship between continuous functions and. Goal. The student will demonstrate the ability to use a problem-solving approach in using Explain the relationship between differentiability and continuity. Relationship Between Differentiability And Continuity: Example Question #1. The function is differentiable at the point. List which of the following statements.
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. 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?
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. Why is a vertical tangent a place where it's hard to define our derivative? Well, remember, our derivative is we're really trying to find our rate of change of y with respect to x, but when you have a vertical tangent, you change your x a very small amount, you have an infinite change in y, either in the positive or the negative direction.
That's one situation where you have no derivative.
Differentiability at a point: graphical (video) | Khan Academy
They tell us where we have a vertical tangent in here, where x is equal to three. F is not differentiable at x equals three because of the vertical tangent.
You might say what about horizontal tangents? No, horizontal tangents are completely fine. Horizontal tangents are places where the derivative is equal to zero. F prime of six is equal to zero. F prime of zero is equal to zero.
What are other scenarios? Well another scenario where you're not gonna have a defined derivative is where the graph is not continuous. We see right over here at x equals negative three, our graph is not continuous. X equals negative three it's not continuous. Those are thee only places where f is not differentiable that they're giving us options on.
We don't know what the graph is doing to the left or the right. These there I guess would be interesting cases. They haven't given us those choices here.
We already said, at x equals 0, the derivative is zero. At x equals six, the derivative is zero. We have a flat tangent. Once again it's defined there as well. Let's do another one of these. Actually, I didn't include, I think that this takes care of this problem, but there's a third scenario in which we have, I'll call it a sharp turn.
This isn't the most mathy definition right over here, but it's easy to recognize. A sharp turn is something like that, or like, well no, that doesn't look too sharp, or like this. The reason why where you have these sharp bends or sharp turns as opposed to something that looks more smooth like that. The reason why we're not differentiable there is as we approach this point, as we approach this point from either side, we have different slopes.
Differentiability at a point: graphical
Notice our slope is positive right over here, as x increases, y is increasing, While the slope is negative here. As you're trying to find the limit of our slope as we approach this point, it's not going to exist because it's different on the left hand side and the right hand side. That's why the sharp turns, I don't see any sharp turns here so it doesn't apply to this example.
Let's do one more examples. Actually this one does have some sharp turns.
This could be interesting. The graph of function f is given to the left right here.