 In this video, I want to explore the different types of discontinuities one runs across in a typical calculus one course. So let's look at a few functions to discuss what type of discontinuities they're going to have. So consider the function f of x equals negative x squared plus 3x plus 4. You times that by x minus 3 and then you're going to divide x minus 3. So as we talked about in the previous slide, this is a rational function. It's a polynomial divided by a polynomial. And rational functions, as we mentioned, are continuous on their domains. So the only place for which we're going to get a discontinuity would be outside its domain, which when you examine the denominator x minus 3, the only thing that makes the denominator go to 0 would be 3 itself. So what type of discontinuity is happening at 3? Because after all, the function is defined, undefined at x equals 3. Well, with this rational function, you will notice there is a factor of x minus 3 on top and bottom, in which case we could, you know, it's tempting to cancel out the x minus 3 and say that, oh, the function equals 1 or minus x squared plus 3x plus 4, which that's sort of true. The function f will agree with that quadratic polynomial at all points in its domain, but 3 is outside the domain. So there's going to be some disagreement there. You're going to see the graph of function f over here. It looks like this parabola negative x squared plus 3x plus 4. The only difference, though, is, of course, at the point x equals 3. The parabola, y equals negative x squared plus 3x plus 4, it's defined for all real numbers because it's a polynomial, but the rational function does have a problem at x equals 3. You'll notice as we illustrate on this graph, this open dot is occurring right here to suggest that the function is defined at all points near this one on the graph, but it's not defined at that point as well. This is an example of what we call a removable discontinuity. A removable discontinuity. Specifically, a removal discontinuity is a continuity where the limit as x approaches c of f of x, this exists, it exists, but this limit as x approaches c of f of x does not equal f of c, right? And because, after all, to be a continuous function, the function has to be defined, the limit has to exist, and the limit and the function have to agree with each other. So if any one of those three properties fails, the function we say is discontinuous at that point. Now, for a removal discontinuity, the limit exists, but it doesn't agree with the function. Now, that could be like in this case because the function f of c is undefined, right? In this case, f of 3 does not exist, right? f of 3 does not exist because when you plug in x equals 3, you get 0 over 0. That's not a number. On the other hand, if we take the limit as x approaches 3 of f of x, this limit does exist. We can actually see that there's a point here. We see that it should be 4, right? That's what the limiting value is going to be. So the limit here would be 4. And where do we get 4? I mean, you can see it from the graph, but basically, if you take this polynomial after you cancel out the x minus 3s, if you plugged in 4 into that, excuse me, if you plugged in 3 into that, you're going to get a 4. Negative 3 squared plus 3 times 3 plus 4, that's going to give you a 4 right there. And so that's how we find this removable discontinuity. The reason we call it a removable discontinuity is because if we were to redefine what happens at f of 3, we could fix it as we could remove the discontinuity just by kind of just filling in the dots right here. Look at that. Now, the discontinuity has been removed because we could redefine f of 3. We could define f of 3 to be 4, and now it fixed the problem. Another type of removable discontinuity is illustrated in this graph. Take the function g of x, which this will be a piecewise function. You'll notice a lot of piecewise functions in our discussions at discontinuities because it's very easy to create discontinuities using a piecewise function. Take g of x to be the function x plus 1 as long as x is itself not 1, but then at 1, x equals 1, we're going to find g to be 3. So g of 1 is equal to 3. So you see that point right here, 1, 3 on the graph. Now, the function's limit suggests that as we get close to x equals 1, the function wants to be 2, right? You see this removed point at 1, 2, but the function's been defined to be at the point 1, 3. Why did we redefine it that way? Well, out of context, it's hard to predict why, but it is a removable discontinuity. We could have redefined this point to be 1, 2. If we moved it, that would fix the problem, right? And so we have this removal discontinuity. g of 1 is equal to 3, but the limit as x approaches 1 of g of x is equal to 2. So the limit exists, the function's defined, but the limit and the function don't agree with each other. Hence, we have this removable discontinuity. As we study limits of difference quotients in this lecture series, these are the types of discontinuities we run across. We get these removable discontinuities where if we simplify the difference quotient, we can then calculate the limit with the simplified form of the function. Now, another type of discontinuity, which we're going to see in this example, is what's commonly referred to as a jump discontinuity. What's a jump discontinuity? Well, let's investigate this picture a little bit more. Let's take the function h of x. It's also a piecewise function. We're going to say h of x is equal to 1 when x is positive, and it's going to equal negative 1 when x is less than or equal to 0, so when it's not positive. And so let's investigate what happens at x equals 0. So you'll notice at the y axis, something funky is happening. There's a jump on this graph. This is a type of step function we see. And so let's let's examine what's going on here. The function is defined at h equals 0. You'll notice that h of 0 is equal to negative 1 by the rule right here. If we look at the left-handed limit, like so as we get closer and closer to x equals 0 from the left, we see that the limit as x approaches 0 from the left of h of x, this will likewise equal negative 1. On the other hand, if we approach 0 from the right, we'll notice that the limit as x approaches 0 plus of h of x here, this is going to equal 1. And so you see there's a disagreement happening here. The left-handed limit is going to be negative 1. The right-handed limit is positive 1. And so a jump discontinuity occurs when the left limit of a function disagrees with the right limit of the function. So if the left limit exists and the right limit exists but they don't agree with each other, we call this a jump discontinuity because they don't match up with one another. Now the function, whatever the function's doing, doesn't matter. We still call it a jump discontinuity. Now in this situation, because the function agrees with the left-handed limit, this is an example of a function which is left continuous. It's not continuous at x equals 0, but it is left continuous because the left-handed limit agrees with it. You see this often with jump discontinuities. A function which has a jump discontinuity could still be left continuous or it could be right continuous if the function agrees with the left-to-right limit. But we could also have redefined this function. Let's say we have an open dot now. I'll use a different color to emphasize this. Maybe we have an open dot here on the bottom, but we filled this one up. This would still be a jump discontinuity, but now it would be right continuous. But hey, we could actually keep that as an open dot. Maybe define the function as something else. Why not? We could say h of 0 is 0 for all we care. In this case, we'd still have a jump discontinuity, but it would be neither left-to-right continuous. In fact, the main difference here between a jump discontinuity and the removable discontinuities we saw earlier is that there's no where we can place the point that fixes the discontinuity. We could define the point down here doesn't work. We could define it right here doesn't work. We could put anywhere else doesn't work. At best, we can make the function be left-to-right continuous, but not both because you have this jump discontinuity. The third type of discontinuity we're going to talk about here is what's called a vertical asymptote. That's going to be illustrated in this example. So consider the function f of x equals 1 over x and consider what happens at x equals 0. And so you'll notice that this function as you get closer and closer to 0, things are going to get very big or very small. So for example, the limit as x approaches 0 from the right of 1 over x, this is approaches will be called positive infinity. On the other hand, if you take the limit as x approaches 0 from the left of 1 over x, you're going to get negative infinity right here. And so we say that a function has a vertical asymptote if the left or right limit. So let's say that x approaches c either from the right or from the left, I don't care. So if x approaches c from the left or from the right, and then the function approaches positive or negative infinity, we refer to that as a vertical asymptote. So we see there's this line that our function is asymptotically approaching right here. And it doesn't matter if they approach in the same direction or not, we'd call this a vertical asymptote. This is a type of discontinuity. Why is it a discontinuity? We'll notice the function is undefined at x equals 0. So it's going to be a discontinuity. And in this case, the limit is approaching plus or minus infinity. Now, so why would this be a discontinuity? Could we change the function so it's not a discontinuity? Like I said, if we'd taken this part and moved upward, we could still get the limit to exist at that moment as x approaches 0 of 1 over x. Let's say we took 1 over x squared, for example, that would be infinity. That still will be a discontinuity, okay? Because if the limit, whether it's the left-handed limit, right-handed limit, or a two-sided limit, if the limit is equal to plus or minus infinity, then there's no chance that the function could agree with the limit. Because our functions have the convention that we only accept real numbers into the domain, and we can only accept real numbers in the range. That is, there's never going to be a point on our function where we're like, oh, f of 7 is equal to infinity. We can't let infinity be a possible output because infinity is not a real number. So if our limits, one-sided or two-sided, ever equal an infinite number, that's automatically going to be a discontinuity because the function cannot output an infinite number either. And so we get what we call this vertical asymptote. As another example, consider g of x equals this piecewise function, 1 over x squared when x is not 0, and x equals 1, excuse me, y equals 1 when x equals 0. In this situation, we still have this vertical asymptote occurring on the y-axis, you can see right there. And when you consider the limit, as kind of mentioned this on the previous slide, when you approach x equals 0 of g of x here, your limit is still going to be infinity, right? So we can tell you what the limit is, but the function g of 0 equals 1, 1 is not infinity, right? So this is a discontinuity and because limits turn out to be infinity here, we call this a vertical asymptote. Another example I want to mention, I don't have the picture prepared, so I'll just kind of show you what's going on right here. Imagine we have some type of like vertical asymptote going on like this. Perhaps our function is doing something like this, right? It approaches infinity, say on the right, but maybe it's actually defined on the left. You get something like this. So in this situation, you would see that the limit as x approaches, let's say this is the line x equals 0. As x equals 0 from the right of our function, we'll call it say f here. f of x does equal infinity, but on the left-hand side, you see that the limit as x approaches 0 from the right of f of x. That's just equal to 0. It agrees with the function evaluation. So this would still be an example called vertical asymptote because even though the function is defined on the vertical asymptote, both of these pictures do that, and even though the function is left continuous as it approaches that x-intercept, which is on the vertical asymptote, because the right-handed limit was infinity, we'd still describe this as a vertical asymptote. And so these three types of discontinuities, removable discontinuities, jump discontinuities and vertical asymptotes, this gives us what we call discontinuities of the first kind. And so these are sort of the benign discontinuities. There are more malignant discontinuities that exist. A prime example would be y equals sine of pi over x. We looked at the topologist sine wave earlier. This function was discontinuous, but it's what we'd say discontinuous of the second kind because unlike these examples of removable jump and discontinuities and vertical asymptotes, the discontinuities of the second kind, the left and right limits don't even exist. For removable discontinuities, the limit exists, which means the left and right hand limits exist. For a jump discontinuity, the left and right limits exist, they just don't agree with each other. And for a vertical asymptote, we see that the left or right limits will probably exist. It's just, I mean, some of them are infinite. Discontinuities of the second kind, the left and right limits don't even exist. And so they're much more malignant. Now, the good news for us in calculus one is we don't really study discontinuities of the second type with the exception of we mentioned them to show that things can get much, much worse than what we're talking about right now. We have enough problems as it is. And so we'll focus on these type of discontinuities for calculus one. Those are the first type.