[MUSIC] Domain of a function. Domain is a set. I'll be using D. It consists, Of all, Permissible values of x. When x takes the values from D, the corresponding y values, Belongs to R, a real number line. And the set of all these values form the range. We explore level curves in the coordinate plane. That works for two dimensional, Sets, two dimensional domains. But what if a function has more than two variables, three and more? In that case we may consider level sets. So level sets, is a similar thing to level curves, but this time what we do, we form sets. So x belongs to domain, and f(x) = y. So we choose some value for y, and we consider all such sets, Such that f(x) = y. Now we start exploring specific properties of the functions, and we start necessarily with Continuous functions. Continuous functions. As we remember from the single variable calculus, the notion of a continuity is heavily based on limits of functions. And that means that, before starting with continuous functions, we'll consider limits. Limits of functions. And let us restrict ourselves [COUGH] for the two dimensional case, n = 2. What it means that a function depending of (x1, x2) has a limit A as (x1, x2) turns to, So let's provide a definition, what it means that number A is a limit of this function of two variables, when x, having coordinates (x1, x2), tends to this particular point. From the start, let me say that not necessarily this point belongs to the domain or the function. If I draw the picture, So this is a fragment of the coordinate plane, and this is the point. So we are interested in the behavior of the function in the neighborhood of this point. Follows, we'll be considering arbitrary sequences which converge to this particular point. And we choose any such a sequence converging to this points and we'll be looking for the values of the function f when the substitute into the arguments there, tenths of the sequence. So first of all, let's suppose that the function is defined in the neighborhood of this point, probably not in the same point. So such neighborhood, by the word neighborhood, we understand this is just a ball centered at this point with some radius. There is a word punctured, a punctured neighborhood. So we would draw this point, because probably this point doesn't belong to the domain. Now, we consider any convergence sequence. So the terms, will, belong to the domain or some bowl, of the radius delta. So that's how we exclude the terms of the side with a point. And let this be a convergence sequence of points. Now, we say that A is the limit, if for any such sequence, the limit to the values, When m turns to infinity equals A. Now, I'll show you how this definition works using some particular example. By the way, quite often, when we deal with a function depending only on two variables like here, we're using a common notation for the x's. x represents the x horizontal axis and y represents the vertical x's. Then, the function we're considering is denoted z, the third letter is used is the function f(x, y). Okay, so let's take some particular function. So the function is defined everywhere except at the origin. So the formula is provided, x times y divided by. Clearly, we exclude the origin because we are not allowed to divide by 0, so this is not 0. Now the question is, whether this function has the limit when the point (x,y) approaches 0 point, whether it has a limit or it does not, and we need to find out. That means that we need to choose any convergent sequence, converging to the origin and substitute the terms of the sequence into the arguments or the function f(x,y). We can use either sequence or we can say that we can approach the origin. Here, this is the origin we're approaching along some curve, or maybe a segment of a straight line. For example, if I choose, let x = y, and this is my t which approaches 0, okay? That means that we have a bisector here. t can be of any sign. So, for example, if t is positive and moving towards the origin from the first quadrants points from the first quadrant approaching the origin. What do we get then? We can do a mental exercise by substituting x and y which equals T into this formula. Then we get f and substituting for x, t and y also t. What do we get then? We get t squared Over 2t squared and that makes one-half. So, the hypothesis is that the function has limit at the origin and it equals one-half. But, what if we change the sequence, so our sequence is a sequence of points which belong to the bisector? We can replace this sequence by another one. For example, we can choose x which approaches 0, but let y = 0. This condition will give us a different sequence of points. Now this time, we chose points from the x axis. And they approach the origin. Now, mental substitution into the formula gives immediately the results. f taken at x and 0 value for y is zero. Now what we see is the violation or this definition. Because how can we accept two different convergent sequences approaching the origin. And for the first sequence, the sequence of the values of the function approaches one-half. For the second sequence, these values approach zero. And here we have two different values for a. And that proves that this function has no limit at the origin. [MUSIC]