 Welcome back to our lecture series Math 12-10, Calculus 1 for students at Southern Utah University. As usual, I'll be your professor today, Dr. Andrew Misseldine. In this lecture 12, we're going to begin our conversation of continuity. This will also extend into lecture 13 in this lecture series. Now, before we define what continuity is properly, I want to actually present you an example of something that is not continuous. That is what we call a discontinuous function. That's because in reality, to appreciate something, we have to see examples of the thing that we're studying, but we have to have to see counter examples. That is what doesn't have that property. It's only in the opposition and in contrast, do we truly understand what something is? We have to see what it's not as well. You don't really know what it's like to feel healthy until you felt unhealthy. You don't know what it feels like to not be hungry until you actually feel hunger. We're going to talk about some discontinuous things first. Imagine the following scenario here. We have a trailer rental firm that charges a flat $8 fee to rent a hitch. Then the trailer itself is rented for $22 a day. This will count for any fraction of the day. If you were to rent the trailer for just an hour and then you return to that same data, it still charge you for the whole day, $22. Let's see if X be the cost function associated to renting a hitch and a trailer for X number of days. If we were to graph this, then we would see something like the picture we see down below, which if you were to rent the trailer for zero days, they would charge you nothing. You didn't take their service. Negative number of days of rental would not make sense. Clearly, negatives will be outside the domain of this function, but any non-negative number would make sense. That does include fractions, decimals, irrational numbers even as well. You potentially could rent a trailer for the square root of two of a day. I mean, that precision might be like, well, that's weird, but that is perfectly reasonable. We could rent a trailer for 1.4 days or what have you, because again, it doesn't matter how much portion of a day you rented, you'll get charged for the whole day if you go past 0.0 in that consideration. For zero days, you could charge nothing. Well, if you were to rent the trailer for any fraction of one day, but any fraction of one day, you're going to get charged $30. Where did the $30 come from? Well, you have the $8 rental fee, the flat fee for the hitch, and you had charged one day for the trailer, so 8 plus 22 is 30. And it doesn't matter if you used half a day, a quarter of a day, 0.99999 day. If you turned it in any time, if you returned the trailer any time before they closed on that day, you would get charged that same fee. But then if you overlap to the second day, whether an hour or the whole day or anywhere in between, you're going to get charged this right here. You're going to get 30 plus 22, which is going to be the $52, whether you see it right here. And so you see there's this type of jump that you went from one day to the second day. There was no sort of nothing in between. There's no way our function could hit anything above 30 below 52. It jumped from $30 to $52. And we see the same thing happen when you hit the third day. If you use any portion of the third day, you're going to get charged another $22 of rental fees there. And so that's going to give you 74 on the third day. And again, any portion of the third day up until the end of the third day. If you rent the trailer any portion of the fourth day, you'll jump again, in which case now we're looking at $96. If you use any portion of the fourth day, then you'll add another $22 and you get $118. And imagine we keep on going, going, going. You're going to get these little segments that look something like this that just repeats over and over and over again. And it'll have this incremental jump of $22 each and every time you do it. Now, when you look at any sector of this graph, you're going to see these segments that look something like the following. It's going to be a flat line that is the Y coordinate will not change. You have a solid dot at the very end, but you have an open dot at the very beginning. Why is that? Well, at the end, so these solid dots represent the end of each day, right? So if you submitted the trailer at the end of day three, then you'd be charged $74. The jump doesn't happen until you go just past, right? You know, so if you take C of three, that's going to equal 74. But if you take C of three, 0, 0, 0, 0, 0, 0, 0, 0, 0, a bunch of zeros, one, you know, that now that you're past three, even by any amount, it's then going to jump up to the $76. And so we have this open dot on the graph here to represent that while we can get arbitrary close to this point, this point's not included on the graph. This is the function value when X equals four. This is the function value when X equals three. This is the function value when X equals two, not this point right here. But let's consider the limits of these things for a moment. Like if we were to consider X equals two right here, if we asked ourself what C of two is, well, according to the rule here, that C of two is going to be $52 because you get charged $8 for the hitch, and then you'll be charged 22 times two for the two days that you've done there, in which case, so C of two is $52. If I asked you what the limit as X approaches two from the right is of our function C of X, we would see something like, okay, as you get closer and closer to, on the right-hand side, you're gonna get closer and closer to this Y coordinate, which is actually on the upper step here, which is gonna be 74. But on the other hand, you're gonna get the limit if you take it from below, that as you approach two from the left, you're gonna get closer and closer and closer to something like this, C of X here, that case is gonna be 52. So what this has to tell us about the limit is that X approaches two of this function C of X, we see that the limit does not exist because the right limit and the left limit don't agree with each other. But you see that for each positive integer in the domain, there's this jump, right? And that jump is actually characterized by this disagreement with the left-hand and the right-hand limits. The left-hand limit agrees with the function, but the right-hand limit agrees with what the next step on this sort of staircase is going to be. So we jump from 52 to 74. And so this is an example of what we call a discontinuity. There's some disagreement between what the function is doing and what the limit expects the function to be doing. A discontinuity would be the absence of continuity. So let's define what that means properly because graphs will sometimes have like this one, jumps or rips or holes, some type of graphical break on the graph. And this will cause the limit to either not exist or to disagree with the function itself. And so discontinuities are about measuring holes in a graph. And so hence a continuous function will be one that doesn't have these type of holes. All right, so let's define what we mean by a continuous function properly. We say that a function F is continuous at some number X equals C. So C is a specific number in the domain of the function F. We say a function's continuous at the point X equals C if the following three things have to hold. So again, these three things have to hold. So the first one is that F of C is defined. This suggests that X equals C is in fact a number in the domain of the function F. So the function has to be defined at the number in order to be continuous. You cannot be continuous outside of your domain. The next thing we need is that the limit as X approaches that number C for F of X must exist. So like the previous example, all of those integer values X equals one, X equals two, X equals three, all of those integer values, the function was defined at them but the limit didn't exist. And that's why it wasn't continuous at those points. So the function has to be defined. The limit has to exist. And then the third kicker, this is the most important part, the limit has to agree with the function. So F of C, this is the function's definition. This is the function assignment. This is what the function is at X equals C. On the other hand, the limit is the expectation. It's the trend. What will we expect the function to do when we look at points near X equals C? And so for a function to be continuous, it needs to do what it's expected to do. So functions which are continuous are not the rebellious teenagers. Mom and dad wants you to grow up to be a lawyer or a doctor and you're like, no, I wanna grow up to be a mathematician. You don't understand me, mom and dad. So when you defy your expectations, that's when you are discontinuous. A continuous function does exactly what it expected to do. And so properly defined, if a function's not continuous at C, we say it's discontinuous at C, or we'd say there's a discontinuity at X equals C. Now, continuous functions get their name because of the following geometric principle. A function is continuous if you could graph the function with one continuous stroke of your pen. That is to say you never have to pick up your pen in order to draw the graph in the XY plane. That doesn't mean you don't have to take pauses, right? There might be like some piecewise functions that fundamentally shift their behavior at these switching numbers. That's okay. It's still as continuous if we can be drawn with no time that you have to pick up your pen. That is one continuous stroke. Now, as continuity is defined using limits, we can also talk about what it means for a function to be continuous from the right, or sometimes we say it's right continuous, to be continuous from the right, well, F of C needs to be defined. The limit as X approaches C from the right has to exist, and the limit as X approaches C from the right has to agree with the function. So it's sort of a weaker form of continuity. Your continuous from the right, if the right handed limit tells you what the function's actually doing. Similarly defined, we can talk about what it means to be continuous from the left or left continuous. What that would mean is the function's defined. The limit as you approach C from the left agrees with the function. So if a function's continuous, it's both right continuous and left continuous, but you can't have a function which is right continuous but not left continuous and vice versa. And of course, you can have a function which is not continuous at all, of course. Going back to this example, this is an example of a function which is gonna be left continuous. It's left continuous. Why is that? Well, notice that when you take any integer value, you're gonna have a function will notice that when you take any integer value, it's gonna be, it's going to be defined. So the function is defined. If you take the left handed limit, the left handed limit exists and agrees with the function. That's why we're left continuous. This is a left continuous function. Now, of course, if you pick any decimal, you know, non-integer and number of the domain right here, then it'll be continuous at that point, right? So the approach from the left and from the right agrees with the function. So this function is left continuous on integers and it's gonna be continuous on, well, non-integers in its domain. And so this kind of step function, as they're sometimes called, they have this property that you're gonna be continuous when you're not near step. And then you might be left or right continuous at the step based upon what the function is doing. You definitely won't be continuous at a step because a discontinuity is what we're seeing when we talk about these steps right here. We can also talk about what it means to be continuous on an open interval. So we say function is continuous on an open interval A to B if it's continuous at each X value in that interval A to B right there. And so similarly, you could talk about being left continuous or right continuous on an interval. The definitions are defined analogously. We can also talk about what it means for a function to be continuous on a closed interval. This one we have to be a little bit more careful about. If you're continuous on a closed interval A to B, that means you'll be continuous on the open interval A to B. You'll be right continuous at X equals A and you'll be left continuous at X equals B. Because the point is if you to the left of X equals A that's outside the interval so we don't care what's going on there. So as you approach X from the right because you can't approach it from the left in this interval, that you don't have to be right continuous the same thing for left continuous there. So that's what it means to be continuous on an interval. Now the good news is that the types of functions we typically run across in calculus, particularly the functions we run across in pre-calculus for the most part are continuous functions. That is they're continuous on their domains because outside your domain you can't be continuous but for most of the functions we run across in calculus that is the pre-calculus functions those are gonna be continuous on their domains. So this would include things like polynomials. Any polynomial function is gonna be continuous for all real numbers because a polynomial function, a polynomial function's domain is all real numbers. A polynomials are continuous for all real, for all numbers in its domain. The significance of this is that since a function's continuous this means that the limit as X approaches C of F of X is equal to F of C. When a function is continuous you can compute its limit just by evaluating the function which is a very, very nice calculation. And as we computed limits earlier in this lecture series oftentimes that's how we did it. We were able to evaluate the function and get the limit. That's what we do with continuous functions because the expectation aka the limit is just what the function is doing the function evaluation there. Polynomials are continuous. Rational functions are likewise continuous functions on their domains where a rational function is a polynomial divided by a polynomial. Now I have to say that they're continuous on their domains which the domain of a rational function will be all numbers that make the denominator non-zero. So if the denominator goes to zero there are potential discontinuities and I guess I should say there are definite discontinuities because if the denominator goes to zero the functions are defined and that's required to be continuous there. So as we start searching for discontinuities of a rational function we're gonna be interested in what numbers make the denominator go to zero there. Let's imagine you have like a square root function f of x equals the square root of x. This function will be continuous on its domain zero to infinity. And like we talked about before that means we're gonna be right continuous at x equals zero. Similar properties can be said for other radical functions cube roots, fourth roots, fifth roots, et cetera. If y equals the cube root of x it'll be continuous for all real numbers. If we take a general power function x to the a right here where x is a variable a is some constant, right? A power function will be continuous on its domain. Now there are some potential concerns, right? Because if you take like y equals one over x that's a rational function so therefore there will be a discontinuity when x equals zero. We talked about the square root of x just a moment ago as well. Because this is a domain concerns there might be some discontinuities associated to that but power functions will be continuous on their domains. Switching over to exponential functions we swap the role of base and exponent there for an exponential function the base is a constant the exponent is the variable. An exponential function so long as the base is positive it'll be continuous for all real numbers because its domain is all real numbers it'll be continuous there. For a logarithm base A so long as the base is positive and not one because otherwise the logarithm is not well defined for logarithms they're gonna be continuous on their domains which with no transformations whatsoever a logarithm's domain will be zero to infinity. Now as you get to the end of the domain at zero you'll notice I did not say that it was continuous here because there's concerns that the logarithm's not defined at zero and in fact as we will see later on a logarithm has a vertical asymptote at that value x equals zero. For trigonometric functions they're continuous on their domains inverse trigonometric functions are continuous on their domains and in fact any function we can create by combining these functions so as we add together subtract, multiply, divide, compose these types of functions we can create new functions from them they will be continuous on their respective domains.