 In the original definition of the derivative, what we defined it as a limit of difference quotients. And so because the derivative is defined with respect to a limit, there's the possibility that that limit doesn't exist. If the limit doesn't exist, then the derivative is undefined, the derivative doesn't exist. So are there conditions for which we can expect the derivative to exist on a function? The answer is yes. And that's what I wanna talk about in this video right here. What conditions are necessary or sufficient to guarantee the existence of the derivative? It turns out there's only three things we have to look for if we were to look at the graph of a function. So the derivative exists when a function F satisfies all of the following three conditions at a point. If these three things happen, that means the function is continuous at that point. So the first one right here is the function has to be continuous. What does continuous mean again? Remember, a function's continuous if it's defined at that point. If the point's undefined, then of course there can't be a tangent line there. Second, the limit at that point has to exist. So to be differentiable, the limit has to exist at the point. But most importantly, when it comes to continuity, continuity means that the limit as X approaches A of F of X is equal to F of A. So a function is continuous at the point X equals A if the limit is equal to the function. That is the function, the function evaluation behaves according to what you would expect it to be. And as we've seen previously, continuous functions are those functions we can draw with one continuous stroke of our pen. All right, so a function does have to be continuous in order to be differentiable. If you're not continuous, then you aren't necessarily differentiable. Let me give you one quick example of that. Let's say we have like a function that is maybe sometimes jump discontinuities going on, maybe something like so, and we get this. Well, if the function was discontinuous, there's a jump discontinuity, as we try to compute the limit, well, the limit, I should say when we compute the derivative, which is the limit of the difference quotient, as it's a limit, we could take the left-handed limit and the right-handed limit. That is, we could look at the left tangent line and the right tangent line. The left tangent line would be something like this, right? We get one type of slope. But then the right tangent line would be something like this and there would be this disagreement. This derivative, this left-handed derivative would want to be negative in this picture, but on the other one, F prime of A wants to be positive. That's a disagreement. The derivative doesn't exist when you have discontinuities. Another concern we have to look out for is, is the function smooth? What do I mean by smooth? I don't mean like a smooth criminal like Michael Jackson or anything like that. Let me show you in contrast what a sharp corner is. Maybe we have some type of corner or cusp on our graph, which when you touch it, ouch, that really hurts. It's a, you know, it's a porcupine quill or something. The issue here is that even though the function's continuous, I drew this with one continuous stroke, the derivative still doesn't exist because the approach from the left gives you one tangent line and the approach from the right gives you a different tangent line with different slopes. And therefore the derivative might not exist. If your graph has a sharp corner, even though it's continuous at that point, we'll see that the derivative will have a jump discontinuity at that point. So we have to watch out for things like that. So to be, to be differentiable, we have to be continuous, we have to be smooth. And the last thing to look out for is that the function cannot have a vertical tangent line. What might that look like? A vertical tangent line can occur on a graph something like this where this function's continuous is completely smooth, right? There's no corners or cusp on it whatsoever. But if you look around right here, there's gonna be this vertical tangent line. If we follow the tangent lines going from the left and from the right, it eventually comes to a vertical line. Why is that a problem? Well, the tangent line slope, if it's vertical, would be undefined. And that's because basically when it comes to rise over one, you have to divide by zero. So to speak, the slope is infinite. The derivative, which is the slope of the tangent line would be that value. So we see that when you have a vertical tangent line, the slope is undefined. So if a function is continuous smooth and has no vertical tangent lines, then it's differentiable. And in fact, a function is differentiable if and only if it's continuous and smooth and has no vertical tangent lines. So if we're looking for places where the function is not differentiable, we look for these three things. If the function has a discontinuity, it's not differentiable at that point. If the graph has a sharp corner or cusp, that means there's no well-defined tangent line. There's no derivative at that point. Also, if it has a vertical tangent line, then there's not gonna be a derivative at that point. If your graph does have a vertical tangent line, this is actually gonna coincide to a vertical asymptote of our derivative. If you have a sharp corner, this is gonna correspond to some type of jump discontinuity on F prime. When F has a discontinuity, it's a little bit harder to predict. And we'll show you an example of what this will look like in just a moment here. But before we continue on here, I wanna actually just give a quick argument that the smoothness makes sense. We talked about that and the vertical tangent lines. But why is continuity so important? Well, that one I actually do wanna prove from a logical point of view. Let's prove the statement that if F is differentiable, then F is continuous. That is continuity is a necessary condition if you're differentiable. In fact, differentiable, you could say is a stronger condition that if you're differentiable, then you must be continuous. Although there do exist things that are continuous but not differentiable. Like take the absolute value V because of the corner. It's not differentiable but it's continuous. So let's assume F is differentiable at the number X equals A. So that's the assumption. Well, what does it mean to be differentiable? It means that the limit or the derivative exists which the derivative is a limit of a difference quotient. So the difference quotient take the limit as X approaches A of F of X minus F of A over X minus A. That limit exists and will equal F prime of A. That's just the abbreviation of it. I also wanna point out that if you take the function X minus A and you take the limit as X approaches A, that'll equal zero just by simple continuity assumptions because you're gonna get A minus A which is equal to zero, basic calculation there. Okay, so come back to this statement. We're gonna write it down here. What we're then gonna do is we're gonna times both sides by the limit, by the limit X minus A as X approaches A. Well, why are we doing that? Well, let's see what happens. On the left hand side, if you take the derivatives limit of difference quotient and multiply it by X minus A, these are two limits that exist by themselves and by limit properties we've seen previously, we can then bring those together to a single limit. It's gonna be the limit of F of X minus F of A over X minus A times X minus A. But the whole point to bring them together is that the X minus A's are gonna cancel out and therefore you just get the limit as X approaches A of F of X minus F of A. That's what happens on the left hand side. Well, on the right hand side, if you take F of F prime of A, any times by the limit of X minus A as X approaches A, well, F prime of A is just a number, right? We don't know what that number is, but it's a number. On the other hand, the limit of X minus A, we can evaluate that and get zero and any number times zero is gonna equal zero. And so we get this important observation right here. The limit of F of X minus F of A as X approaches A is gonna equal zero. I should also mention that if you take the limit as X approaches A of F of X minus F of A, that limit, well, we just showed that that limit exists. The other thing I wanted to mention was that if we take the limit of F of A as X approaches A, that's gonna be F of A. Because like I mentioned earlier, these F primes of A and these F of A, these are just a constant. So if you take the limit of a constant function, the limit will just be that constant value. That's important because we're gonna take this limit and we're gonna add it with this limit. So if you take the limit of F of X minus F of A and add to it the limit of F of A, that's perfectly appropriate by limit laws. But what's good for the goose is good for the gander. That is to say, whatever we do with the left-hand side of the equation, we have to do it to the right-hand side of the equation as well. So take this equation which we established already and add to both sides of the equation the limit of F of A. On the right-hand side, because these limits exist by themselves, we can combine them together. And so you're gonna get the limit of F of X minus F of A plus F of X. The F of A's cancel and you get the limit of F of X. The right-hand side, well, the limit of F of A, like we said earlier, is just F of A. And so when we then get the limit of F of X is equal to F of A, which then proves that the function is continuous if it was differentiable.