[MUSIC] In this video, we'll discuss and illustrate the important notion of continuity of a function, what can go wrong when discontinuities occur, and how we handle them. Including examples when we can fix things using removable discontinuities. A function is said to be continuous if a graph can be drawn, at least in principle, without lifting your pen off the paper. For example, the straight line y = x in this diagram was drawn without lifting the pen. The line, of course, goes on forever. So one has to imagine in principle being able to draw as much as we like, back and forth, as far as we like, without ever needing to lift the pen. In fact, this works for any graph of a linear function. Here's a horizontal line which is the graph of y = k for some constant k. Which again, we can imagine drawing without lifting our pen off the paper. Here's the parabola y = x squared, also continuous, and y = sine x, which wriggles back and forth, forever but continuously. And y = e to the x and y = the natural logarithm of x, all continuous functions. When we imagine drawing without lifting our pen, there are no holes or gaps. And the limiting values precisely fill in where your pen is supposed to move to, captured by the statement, the limits of f(x) as x approaches a is the value of the function a, namely f(a). And this has to work for every input x equal to a in the domain of the function. There are no surprise jumps or leaps, and you can think of any limiting behavior of the function as coinciding with the actual value of the function. There are two main obstructions to continuity in practice. Firstly, we must consider denominators of fractions. We're not allowed to divide by zero, and so have to avoid values that cause the denominator of a fraction to be zero. The second main problem is when a function is defined piecewise, which means that there are two more pieces in the description of the rule for the function. Leading to two or more pieces of the curve which have to match and join up with the endpoints, if we are to imagine drawing the combined curve without lifting a pen off the paper. Here's an example of a piecewise function that happens to be continuous. Consider the usual magnitude function, f(x) = the magnitude or absolute value of x. In fact defined in two pieces, as x, if x is greater than or equal to 0, and minus x if x is less than 0. The graph is in a V-shape, formed by taking the line y equals x in the first quadrant in the plane, and combining it with y equals -x in the second quadrant. The two pieces of the curve join up, matching perfectly at the origin. You can imagine drawing the V-shape without lifting your pen off the paper. By contrast, consider the following example, which turns out to be discontinuous. By which we mean that we'll be unable to draw the graph without lifting our pen off the paper. For this, we modify the previous example, and consider the function with rule f(x) = x divided by the magnitude of x. Which again can be describe in pieces. When x is positive, the magnitude of x is x itself, so dividing x by x gives the value 1. When x is negative, the magnitude of x is -x, so dividing x by -x gives the value -1. The rule is undefined for x equals 0. Here's the graph of y = f(x) which consists of two horizontal lines. One at height y equals 1 for x greater than 0, and another at height y equals -1 for x less than 0. Note that these two lines get arbitrarily close to the y-axis, but don't actually meet it. You can think of there being two holes on the y-axis, and y equals 1 and y equals -1. It won't make the overall curve continuous by filling either of these holes. For x approaching 0 from above, from the positive right-hand size, the y values are stuck at plus 1, so the limit from the right is plus 1. By contrast, for x approaching 0 from below, from the negative left-hand side, the y values are stuck at -1, so the limit from the left is -1. We have these contrasting one-sided limiting behaviors for x over the magnitude of x. We have a limit existing from the right, as x tend to 0 from above, namely plus 1. And a limit existing from the left as x tend to 0 from below, namely -1. The overall two-sided limit does not exist if we allow x to approach 0 from both sides. There's no hope of filling in the holes to make the curve continuous so that you can draw it without lifting your pen off the paper. Recall that rational functions are ratios of polynomials, of the general form y = f(x) = p(x) divided by q(x), where p(x) and q(x) are polynomials. Problematic behavior occurs when denominators become zero, often but not always leading to asymptotes. Recall two contrasting examples from an earlier video, f(x) = x squared -1 divided by x-1, and g(x) = x squared -2 divided by x-1. Both of which would have 0 denominator if x = 1. So that the input x = 1 is problematic, but in quite different ways. Recall, we produced the graph of f, which turns out to be a line with a hole at the point (1, 2). And the graph of g, which turns out to have an oblique asymptote, and a vertical asymptote passing through the x-axis at x = 1. In the case of the function f, the rule becomes f(x) = x + 1 for x not equal to 1, producing a line with just one point missing. If we fill in the hole, then we get the full, continuous curve. We say that the discontinuity is removable. By contrast, if we look at the graph of g, no matter where we travel on the vertical asymptote corresponding to x = 1, we cannot find an input that will join up to two branches of the curve. They always remain separated. We say that the discontinuity is not removable. I'd like to finish by describing one of the most famous and important examples of all that has a removable discontinuity. Recall that last time we carefully demonstrated using the Squeeze Limit law, that the limit of sine x over x is equal to 1 as x tends to 0. Let's consider the function f(x) = sine x over x, with domain the set of all nonzero real numbers. The rule is not defined for x = 0 because of x in the denominator, but it's defined for all other real numbers x. Here is the graph of f. Note that there is a hole for x = 0, because the rule is undefined there. Because of the wavy periodic behavior of sine x, we expect the curve to undulate forever to the right and left. However, the amplitudes of the undulations decrease the further we're away from the origin, as the denominator x gets larger and larger. Multiplying by 1 on x has some kind of dampening effect on sine x. The graph also has perfect reflectional symmetry about the y-axis. It's an example of an even function, a concept that we'll talk about and use later in the course. As x approaches 0 from either side, y = sine x over x approaches 1. Because the limit exists as x approaches 0, we can fill in the hole, and then adjust the rule for the function. Which remains sine x over x for nonzero x. But now takes the value 1 in the special case when the input x is equal to 0. The discontinuity turns out to be removable, and we produce this beautiful, continuous, smooth curve. It's a famous curve that has many application in areas of physics and engineering, such as signal analysis, acoustics, optics, and spectroscopy. Today we moved along at a very fact rate, and included quite an advanced discussion about limits in the context now of continuity. We discussed the notion of a continuous function, whose graph we may think of as being drawn without lifting the pen off the paper. A property that can be defined in terms of limits, namely that the limiting behavior of the curve at any particular point actually takes the limiting value. We gave contrasting examples, some with a removable discontinuity at a particular point where the limit exists, including the famous sine x on x curve. In cases where it's not possible to remove the discontinuity, including examples such as where the left and right limits exist but are not equal, and where there's a vertical asymptote for a rational function. Please read the notes, and when you're ready, please attempt the exercises. Thank you very much for watching, and I look forward to seeing you again soon. [MUSIC]
