 Welcome to a screencast about continuity. So the question of the day is, looking at a function, how can you tell if it is continuous at a certain point? So recall from the text that the definition of a function being continuous at a point x equals a has to satisfy three conditions. f of x, the function, has to have a limit as x approaches a. f of x has to be defined at x equals a. And those two values have to be the same. So the limit as x approaches a of f of x has to equal f of a. All right, so let's take a look at a function. And here we're going to start with a graph. I always think looking at stuff graphically is the best thing to do. So as you read this graph left to right, we can assume that it kind of continues even as you go off the page, or even if you just want to look at negative 6 to 8, as it turns out, for the window here. So you want to look at the graph and think, all right, so if I had is my function defined, is my limit defined, and are those the same values? So for most of the functions that you're going to be able to draw and think about, they are going to be continuous. But you notice the function that I drew here, and by the way, I did this on geogra by myself the first time I've ever done a graph and look at the funkiness I created here. So anyway, I came up with a graph, and it's got some issues as you read the graph from left to right. So that's what we want to discuss. So what is happening at those issues? So the first place that I see an issue at is right here, at x equals negative 1, because what do you notice is happening at x equals negative 1? Well, the functions, I guess the function is defined, because it's got a point up here. The limit is defined, because as you approach negative 1 from the left, and as you approach negative 1 from the right, it seems to have the same value. But are these two pieces the same? And the answer is no. So the function is not continuous at negative 1, because the function value does not equal the limit value. And as you were drawing this graph from left to right, you'd have to make a hole, pick up your pencil, put the dot in up here, go back, finish your graph. So it's kind of a good intuitive feel then as well. So why don't you pause the video for just a second and see if you can identify any other points on the graph where the function is not continuous. OK, well, hopefully you decided 1 also causes an issue. And what could we say is wrong with this one? Well, of our three parts that we have, is the function defined? Yes, I see a closed circle up here at 4. Is the limit defined? I don't know. So as we approach one from the left, it looks like our y values are approaching negative 1. As we approach from the right, it looks like they're approaching 4. So no, the limit does not exist at 1. So 1 is a problem. Again, if you were drawing this graph, you would have to draw the straight line, do the open circle, hop up here, and then go ahead and draw in this parabola-like shape. So obviously, something funky is going on here. OK, another funky part I hope you noticed is at x equals 6. So is the function defined there? Thinking back to your algebra days, what is this little dotted line I have in here mean? That means there's an asymptote. So obviously, the function is not defined. OK, that definitely falls apart. If you didn't notice that, is our limit defined? Well, as we approach 6 from the left, this graph is shooting up. But as we approach 6 from the right, the graph is shooting down. So obviously, the limit's not going to exist either. So to recap, you have problems if you have a hole in the graph, which is what is happening in negative 1. If you have a jump in the graph, which is what's happening here at x equals 1, or if you have an asymptote. So that's what's happening here at x equals 6. There are other places where you could have problems with continuity, but these are definitely the main three. OK, so now that we've looked at a graph, let's take a look at a piecewise function for those of you who are more algebraically inclined. So for this one, I define the function g of x to be the function 3 if x is less than or equal to 0. And I define the function to be x squared minus 1 if x, I don't know why I wrote it this way, is 0 less than x. This is the same thing as saying x is greater than 0. Just fuddled me for a second there. All right, so thinking about things as you're drawing this, so 3, that's a nice function. It's got a flat line to it, but it's not going to have any problems. x squared minus 1 is a nice parabola-like function. Again, not going to have any problems. But where does this function going to have problems at? Well, we put those two pieces together. So at x equals 0 is going to be our issue. So we've got to figure out what's going on the left side of 0, what's going on the right side of 0. Does a limit exist? So we can answer the function question, obviously, because I have less than or equal to right in here. Let me erase this line that I just put in. So here you can tell that definitely because of the way this is defined, the function exists. So g of 0 is equal to 3. So now we have to figure out, does the limit exist? And then if it does, are those two values going to actually be the same thing? So less than 0 means that the limit is less than or the limit as x approaches 0 from the left of our function is going to be 3. There's no other way. This is a flat line coming to 0. Nothing else is going to happen. Let's take a limit, though, look at the limit as we approach 0 from the right of our function. So that means we're going to need the second part of this piece. So if we were to plug in 0 into that function, we'd end up with negative 1. Are these two values the same? No. So the limit does not exist. But you notice that I asked about the continuity of this piecewise function. So we could say that this function is continuous everywhere except at x equals 0 because that's where this graph has the jump at. If we were to graph it, that's where this function has the jump at. But everywhere else, it's pretty well-baved. Thank you for watching.