Analyzing functions for discontinuities
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Types of discontinuities
- [Instructor] What we're going to do in this video is talk about the various types of discontinuities that you've probably seen when you took algebra, or precalculus, but then relate it to our understanding of both two-sided limits and one-sided limits. So let's first review the classification of discontinuities. So here on the left, you see that this curve looks just like y equals x squared, until we get to x equals three. And instead of it being three squared, at this point you have this opening, and instead the function at three is defined at four. But then it keeps going and it looks just like y equals x squared. This is known as a point, or a removable, discontinuity. And it's called that for obvious reasons. You're discontinuous at that point. You might imagine defining or redefining the function at that point so it is continuous, so that this discontinuity is removable. But then how does this relate to our definition of continuity? Well, let's remind ourselves our definition of continuity. We say f is continuous, continuous, if and only if, or let me write f continuous at x equals c, if and only if the limit as x approaches c of f of x is equal to the actual value of the function when x is equal to c. So why does this one fail? Well, the two-sided limit actually exists. You could find, if we say c in this case is three, the limit as x approaches three of f of x, it looks like, and if you graphically inspect this, and I actually know this is the graph of y equals x squared, except at that discontinuity right over there, this is equal to nine. But the issue is, the way this graph has been depicted, this is not the same thing as the value of the function. This function f of three, the way it's been graphed, f of three is equal to four. So this is a situation where this two-sided limit exists, but it's not equal to the value of that function. You might see other circumstances where the function isn't even defined there, so that isn't even there. And so, once again, the limit might exist, but the function might not be defined there. So, in either case, you aren't going to meet this criteria for continuity. And so that's how a point or removable discontinuity, why it is discontinuous with regards to our limit definition of continuity. So now let's look at this second example. If we looked at our intuitive continuity test, if we would just try to trace this thing, we see that once we get to x equals two, I have to pick up my pencil to keep tracing it. And so that's a pretty good sign that we are discontinuous. We see that over here as well. If I'm tracing this function, I gotta pick up my pencil to, I can't go to that point. I have to jump down here, and then keep going right over there. So in either case I have to pick up my pencil. And so, intuitively, it is discontinuous. But this particular type of discontinuity, where I am making a jump from one point, and then I'm making a jump down here to continue, it is intuitively called a jump discontinuity, discontinuity. And this is, of course, a point removable discontinuity. And so how does this relate to limits? Well, here, the left and right-handed limits exist, but they're not the same thing, so you don't have a two-sided limit. So, for example, for this one in particular, for all the x-values up to and including x equals two, this is the graph of y equals x squared. And then for x greater than two, it's the graph of square root of x. So in this scenario, if you were to take the limit of f of x as x approaches two from the left, from the left, this is going to be equal to four, you're approaching this value. And that actually is the value of the function. But if you were to take the limit as x approaches two from the right of f of x, what is that going to be equal to? Well, approaching from the right, this is actually the square root of x, so it's approaching the square root of two. You wouldn't know it's the square root of two just by looking at this. I know that, just because when I, when I went on to Desmos and defined the function, that's the function that I used. But it's clear even visually that you're approaching two different values when you approach from the left than when you approach from the right. So even though the one-sided limits exist, they're not approaching the same thing, so the two-sided limit doesn't exist. And if the two-sided limit doesn't exist, it for sure cannot be equal to the value of the function there, even if the function is defined. So that's why the jump discontinuity is failing this test. Now, once again, it's intuitive. You're seeing that, hey, I gotta jump, I gotta pick up my pencil. These two things are not connected to each other. Finally, what you see here is, when you learned precalculus, often known as an asymptotic discontinuity, asymptotic, asymptotic discontinuity, discontinuity. And, intuitively, you have an asymptote here. It's a vertical asymptote at x equals two. If I were to try to trace the graph from the left, I would just keep on going. In fact, I would be doing it forever, 'cause it's, it would be infinitely, it would be unbounded as I get closer and closer to x equals two from the left. And if try to get to x equals two from the right, once again I get unbounded up. But even if I could, and when I say it's unbounded, it goes to infinity, so it's actually impossible in a mortal's lifespan to try to trace the whole thing. But you get the sense that, hey, there's no way that I could draw from here to here without picking up my pencil. And if you wanna relate it to our notion of limits, it's that both the left and right-handed limits are unbounded, so they officially don't exist. So if they don't exist, then we can't meet these conditions. So if I were to say, the limit as x approaches two from the left-hand side of f of x, we can see that it goes unbounded in the negative direction. You might sometimes see someone write something like this, negative infinity. But that's a little handwavy with the math. The more correct way to say it is it's just unbounded, unbounded. And, likewise, if we thought about the limit as x approaches two from the right of f of x, it is now unbounded towards positive infinity. So this, once again, this is also, this is also unbounded. And because it's unbounded and this limit does not exist, it can't meet these conditions. And so we are going to be discontinuous. So this is a point or removable discontinuity, jump discontinuity, I'm jumping, and then we have these asymptotes, a vertical asymptote. This is an asymptotic discontinuity.