Hi everyone. Welcome to our lecture on Continuity. Most of you have seen continuous functions before, or heard of the idea continuity, but if I asked them what does it really mean when we want to study harder functions, the best answer they're able tell me is, it's the graph of the function you draw without picking up your pencil. That's a little tough. It's a little bit of a hand wavy definition. It's not really rigorous, so let's use our new language of calculus so far, and actually do it for real. We say a function f(x) is continuous at a number a, specific number r, so real number a, and here's the definition, If, I'm going to put it down here, the limit, as x goes to a of f(x), equals f(a). Once again, this is another definition that we need limits to properly define. As we get into harder functions, more difficult functions that model the real world, we're going to need this definition, and we need to understand limits, and in particular, we need to evaluate limits to calculate this left hand side. A couple of things. It's not exactly explicitly listed, but assumes the following things. One, that you can evaluate, f(a). So f(a) exists. What does that mean? We only talk about continuity, if a is inside the domain of f. It doesn't make sense to talk about continuity at a point if the point is not in the domain. For example, if I have the function f(x) equals one over x, so remember what this graph looks like. It is something like this, and I say to you, let's talk about a equals zero, and is the function f(x) continuous at zero? This question actually doesn't make sense. We would say, a is on the domain, continuity doesn't make sense. It's not continuous at this point. You can't evaluate the function at zero, so clearly the right hand side is not a number, which means I can't even compute the limit. We talk about continuity at values of a inside the domain. The other part that we assume as well, not only does the right side exists, that the left side, the limit, as x approaches a of f(x), exists as well. We saw before that it's possible for limits to not exist. Again at zero, here in the example, one over x doesn't exist. So we would not have a number to compare with the point evaluated at the function, so this one over here at zero, a lot of discontinuity is occurring. Lastly finally, that the limit is equal, that the limit of f(a) is equal to f(a). Limit as x approaches a of f(x) is equal to f(a). If one of these three things are broken, if the function is not continuous, so if f(x) is not continuous, cts is my abbreviation for continuous, at a point x equals a, then we say that f(x) is discontinuous. Again, think of functions that are, think of functions that are not. Here's a new definition, and really understand how it works based on the definition. We're going to see difficult functions where the graph gets tricky, so you need to revert back to the definition to answer the questions. Just some examples, if I have a graph with a nice hole in it, and let's define it outside the hole up here, so here is f(x), put it back, and call this 0.1. If I asked you, is f(x) continuous at x equals one, what would you say? Now again, you can see this is a baby example. I can't draw the graph without picking up my pencil. But let's talk about why it's actually like, if you write that on the test or homework, you're going to lose some points. Let's see why not. Let's give this thing some numbers here, one, and two. The limit as x approaches one of the function, so where does the function want to go, is equal to one, but the function evaluated at one is equal to two, at this dot above the open circle. Since they are not equal, therefore the function is not continuous at x equals one. This is a nice one here. We saw, why not. Let's talk about the greatest integer function. This is another one that doesn't come up a lot, but it's good to have it ready as needed. Here's a function, has closed circle on the left, and open circle on the right, and it walks up, it's a staircase graph, every integer does its thing, and it goes down to the negatives as well. Let's just pick x equals one, so that's like right here, this function is discontinuous. The function itself at f of one is equal to one. If I plug in the function of around down at one, I get one, but the limit as x approaches one. What happens is I approach to the right, as I approach from the left of the greatest integer function this does not exist. Since the limit doesn't exist and it's not equal to one in particular, this is not continuous at x equals one. Other examples that may come up, how about f of x equals e to the x. So this graph, we should know exponential growth. This one here I can draw without picking up my pencil but of course, for any value, the value of the limit where the function wants to go was equal to the function evaluated at that point. This function is continuous, so we say e to the x is continuous at all x in r. So it's continuous on the real line. Not every function has to have a discontinuity, remember that how you have a discontinuity is different, we call these holes in the graph removable discontinuity. We call these when you have a staircase or something like this when you jump, they're called jump discontinuities, so they have different names. This function here, u to the x, we'd say it's a continuous function. Again, when you learn a new idea, when you learn a new definition, have examples of something that is the definition and something that is not. When you have a function that is continuous on an entire interval, we say the whole function is continuous when it's on it's domain. This example here we'd say, if you wanted to write this out as a sentence, continuous on r. If I hand you an interval, if I hand you a set, if I tell you I'm continuous on that interval, it means that I'm continuous at every number inside that interval. So just like limits, you can also talk about one sided continuity, and so let's put those definitions down here as well. We say definition, we say that f of x is continuous from the right, and there will be a similar definition for continuous from the left if the one-sided limit, the limit as x goes to a number from the right of f of x is equal to f of a. Similarly, if I did to continuous from the left, very very similar, if the one-sided limit as x approaches a from the left, f of x equals f of a. You can talk about one-sided continuity and just as an example, let's just take f of x equals the greatest integer function again, draw it's graph closed circle to open, closed circle to open, closed circle to open, I will just do this one. Here we saw before that this function is not continuous at x equals one. The limit at one does not exist, however, this graph is great because it's the prototype, it's the example for one-sided limits, so where is it continuous from the right, is it continuous from the left anywhere? But let's look at the limit as x approaches one from the right, where does the function want to go as I approach from the right? You're a little bug, you're moving this way, the function wants to go right to one, and what is the function evaluated at f of one, rounding down from one equals to one. Now they're the same, so the two side of them fails, but the one-sided limit actually works and they're equal, so put these two together and you can say that the staircase function, the greatest integer function, the floor function, whatever you want to call it, is in fact continuous from the right. The main idea here is, don't tell me something is by drawing me a picture, know proofs by pictures always revert back to the definition. Always revert back to the definition, so this is a definition you must memorize, it'll come in handy as you go through some of your examples. Next up is property of continuous function. Now because continuous functions involve limits, and limits are so nice and how they interact and behave with functions, we have a theorem that follows. If f of x and g of x are two functions that are both continuous at a point x equals a, then the following functions are also continuous at x equals a. So not only are two, so assume you have two functions that are continuous, if you have two functions, if you add their functions together, this or subtract, which is just the addition of negative numbers, they're negative functions, you will get back a continuous function. That is because limits, of course, distribute over addition and subtraction. If you take any constant and multiply it by a continuous function, you will get back a continuous function. Why? Because the property of being continuous comes from the fact that you have a nice limit. Limits don't care about constants in front so neither does continuity. Limits distribute over products so of course, if I have two continuous functions, their product is continuous and same with division. Of course all the caveats for division apply assuming that g of x is not equal to zero. These all follow from, and this is the good chance to go back and look how they follow immediately. Follow from the limit laws. So basically, if you are continuous, your sum, your difference, your product of times up scalar, they are all continuous as well. So the continuous functions play nicely together. That brings us to some, I guess, results or observations. Any polynomial. But how do you make a polynomial? You just take a continuous function, you add, you throw [inaudible] in front, you basically build it using these laws. Any polynomial is continuous everywhere, so in particular on the real numbers. If you give me any polynomial; degree five, degree 10, degree a million, who cares? Its graph will be drawn picking up its pencil. It will have the limit found by evaluating at any function. Once you talk about polynomials, the next class of functions to talk about is rational functions. Any rational function, remember rational function is a quotient of polynomials. Like p of x over q of x, any rational function is continuous on its domain. We saw the example before like 1 over x and the domain remembers just where q of x is not zero. Any value of x, as long as the denominator is not zero, the rational function will be continuous. We saw this example with 1 over x. Good. So let's do some examples here. More examples here of what are some nice continuous functions. Now, sine of x and cosine of x, these are not polynomials we should say. Some students like trying to tell me they are polynomials, but of course that is not true. They are good old trig functions, they are not polynomials. If you graph sine of x, it does this little thing, it waves and oscillates between 1 and negative 1. This is a beautiful continuous function. Continuous where? Continuous on its domain, continuous on all reals. Cosine of x, it's another one. This is the one that has even symmetry. It's symmetric about the y-axis, it does its thing. This is also continuous on all of r, on all the real numbers. Then tangent function, which is a friendly reminder, is sine of x over cosine of x. This is continuous except where the denominator is zero. These are just some graphs because maybe we view it as a reminder of some of the graphs. Pi over 2. Negative pi over 2. As long as your denominator is not zero, then you are continuous. Basically, wherever cosine is not zero. So that happens at Pi over 2 and 3 Pi over 2 and it has infinitely many discontinuities. It goes on forever and ever. These are all graphs that you should know. Treat this little section just as a reminder of what some of the graphs look like and what the continuity behavior of these graphs are. Just more examples that are not polynomials as well. I will draw the arrow this way now. How about the exponential function that we are going to see a lot. We saw this before. This is continuous on all of r, goes right through 1. This is continuous on the real numbers. Another nice reason, another good property of the exponential function, why we like it so much, among many other reasons. It's inverse, the natural logarithm. The inverse natural log goes low to high. It is also continuous on the real numbers as well. These are just lots of graphs that come up. You have been working with continuous functions, maybe probably knowing that they are, but the key takeaway here is, this has a real definition. The mistake they do the thing not to tell me is, "I drew it without picking up my pencil." Because we are going to see functions that get really complicated, where that does not even make any sense. Right now for these simple examples, these examples that are common, these pre-calc examples, know their domains. Know the continuity of these functions. Know if they have asymptotes and this thing. All right, sounds good. We will stop here and then we will talk more about other examples in the next section. All right, memorize your definitions and again as always have examples of something that is and something that is not. All right, see you next time.
