Connecting differentiability and continuity: determining when derivatives do and do not exist
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Differentiability and continuity
- [Instructor] What we're going to do in this video is explore the notion of differentiability at a point. And that is just a fancy way of saying 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 to the limit as X approaches Z of F of X, minus F of C, over X minus C. And at first when you see this formula, 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 gets closer and closer to zero? And we talk about that in other videos. 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 into the proof direction. But this is more to get an intuition. And so the first claim that I'm going to make is if F is differentiable, at X equals C, at X equals C, then F is continuous at X equals C. So I'm saying if we know it's differentiable, if we can find this limit, 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 will not be differentiable. If F not continuous at X equals C, then F is not differentiable, differentiable at X is equal to C. 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? Well, for this point, X comma F of X, 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 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. I've just given you good arguments for when you're not continuous, you're not going to be differentiable, but can we make another claim that if you are continuous, then you definitely will be differentiable? Well, it turns out that there are for sure many functions, an infinite number of functions, that can be continuous at C, but not differentiable. So for example, this could be an absolute value function. It doesn't have to be an absolute value function, but this could be Y is equal to the absolute value of X minus C. And why is this one not differentiable at C? Well, think about what's happening. Think about this expression. Remember, this expression all it's doing is calculating the slope between the point X comma F of X and the point C comma F of C. So if X is, say, out here, this is X comma F of X, it's going to be calculated, so if we take the limit as X approaches C from the left, we'll be looking at this slope. And as we get closer, we'll be looking at this slope which is actually going to be the same. In this case it would be a negative one. So as X approaches C from the left, this expression would be negative one. But as X approaches C from the right, this expression is going to be one. The slope of the line that connects these points is one. The slope of the line that connects these points is one. So the limit of this expression, or I would say the value of this expression, is approaching two different values as X approaches C from the left or the right. From the left, it's approaching negative one, or it's constantly negative one and so it's approaching negative one, you could say. And from the right, it's one, and it's approaching one the entire time. And so we know if you're approaching two different values from on the left side or the right side of the limit, then this limit will not exist. So here, this is not, not differentiable. And even intuitively, we think of the derivative as the slope of the tangent line. And you could actually draw an infinite number of tangent lines here. That's one way to think about it. You could say, well, maybe this is the tangent line right over there, but then why can't I make something like this the tangent line? That only intersects at the point C comma zero. And then you could keep doing things like that. Why can't that be the tangent line? And you could go on and on and on. So the big takeaways here, at least intuitively, in a future video I'm going to prove to you that if F is differentiable at C that it is continuous at C, which can also be interpreted as that if you're not continuous at C, then you're not gonna be differentiable. These two examples will hopefully give you some intuition for that. But it's not the case that if something is continuous that it has to be differentiable. It oftentimes will be differentiable, but it doesn't have to be differentiable, and this absolute value function is an example of a continuous function at C, but it is not differentiable at C.
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