 nice theorem, which says a function is of bound. We said every monotone function is a bounded variation. It says if a function is a bounded variation, if and only if it is a difference of two monotonically increasing functions. This is the only way functions of bounded variation can arise. So, this is a theorem due to Jordan. So, it characterizes what are functions of bounded variation. That is why, so only to state this result that every monotone function is a bounded variation and conversely bounded variation is a difference of two monotone. I brought in all those properties. So, just understand what is the definition. Some examples remaining properties just go through once and so that we are exposed to those properties. So, this is about monotone function. I think let us start looking at, so what kind of functions we looked at? We looked at functions defined on real line or Rn. We looked at limits of such functions and continuity of these functions. Then, we looked at what are uniformly continuous functions. All these motions were defined on Rn. You can have continuity, you can have notion of, but let me say something about continuity on Rn. So, continuity. We proved some properties of continuous functions. Namely, if a domain is compact, then the range also is compact. Domain is connected, then the range is also connected. But, there is some basic differences which arise. Let us look at that. So, to understand, let me take n equal to 2 because that will highlight everything. So, f is a function defined in a domain in R2 taking values in R. So, x comma y goes to f of x comma y. So, what was f continuous at a point, say a, b? So, the definition was for every sequence x n, y n belonging to D, converging in D, x n, y n converging to a, b should imply f of x n, y n converges to f of a, b. The limit should exist and should be equal to the value of the function, that is what essentially. In terms of neighborhoods, this was equivalent to saying for every epsilon bigger than 0, there is a delta bigger than 0 such that whenever the distance between x, y minus a, b is less than delta, that should imply f of x, y minus f of a, b is less than epsilon. That was the neighborhood epsilon delta definition correspondingly. Whenever x, y is closed by distance delta, f of x, y minus f of a, b is closed by epsilon. For every epsilon, there is a delta. So, this we had all seen, this was a continuity and worked very well. Let us, here is a note or a remark. See, very often one would like to, f is a function of two variables f, D contained in r tau to r, x, y goes to f of x, y. One can fix, say y naught belonging to D and consider, so y naught is fixed, second coordinate is fixed, first coordinate is varying. So, this goes to f of x, y naught. In the domain, I fix one of the coordinates and let the other coordinate vary. So, what it looks like, let me draw a picture of the domain. So, this is, this is the domain D, that is a subset of r tau. So, this is, this is y and that is x. So, we have fixed y naught. So, that means what? So, here is y naught, it is fixed, x is varying and f of that is defined. So, we are looking at all x, y naught in the domain. So, what are all, in the picture, what is all such things? So, you are looking at all x, so that this is, so this you are looking at this line. So, that is all x, so that x, y naught is in the domain and you are looking at the image. So, you are looking at this is a function of one variable, one variable. X is fixed, sorry, y is fixed at y 0 and you are looking at the function of x. And you can also do the other way round. Instead of x naught, you can fix y naught, you can fix some x naught. So, you can fix some x naught here, x naught and look at what happens when you go around this line. So, one can also fix x naught, this does not make sense, x naught in D. D is a function of two variables. So, fix x naught and fix x naught in R and consider x naught comma y belonging to D. So, look at the function of x naught comma y goes to f of x naught comma y. So, along that part of the domain. So, these are called, you are fixing one of the coordinates. So, for a function of two variables, you can fix either of the variables, either of the coordinates, you get a function of one variable. So, two functions of one variable, these are called coordinate functions, are called coordinate, fixing one of the coordinates. So, each is a function of a single variable. Now, you can, one can ask a question, if I am given a function of two variables, which is continuous in the domain D, what are these coordinate functions? They will same function f, but the domain is restricted along that lines. So, is coordinate functions continuous? So, here is a observation. So, note f continuous implies both coordinate functions are also continuous, both coordinate functions are also continuous. Is it clear why it is so? Obviously, because, so let us consider one. So, fix y naught, then we want to look at x comma y naught going to f of x comma y naught. I want to check whether this function is continuous or not, is continuous. Because, how do I check? I can do it by sequences, if x n converging to x naught, then what happens to x n comma y naught is 0, that converges to x naught x n comma y 0 as a sequence in R 2, which implies by continuity of f, f of x n y 0 converges to f of x 0 y 0. So, it converges to x 0 y 0 and that converges to x n. So, hence, so that is okay. So, the coordinate functions are continuous. So, in fact, what we are saying is the following. See, if you want to look at the picture, so this is the point x 0, what we have fixed? We have fixed y 0. So, this is y 0 fixed here. So, if I take a sequence x n converging to x 0, then what happens? So, this is x 0, then the corresponding sequence in R 2 will converge to x 0 y 0. So, the function is continuous. So, it will give f of x n y 0 converges to, so that is what we are saying here. So, this is one coordinate. This goes to in the domain now, this is in the domain, x n y 0 goes to, x n y 0 in the domain, f is continuous. So, that implies this. So, this is not one. Can I say the converse is true? f is a function, so that each coordinate function is continuous. Can I say the function of two variables itself is continuous? So, let us ask the question. If both coordinate functions are continuous, can we say f itself is continuous? The answer is no, no in general. That means what? You can construct functions such that their coordinate functions are continuous, but the function is not continuous. So, try, let me for the time being leave it as an exercise. In two variables, when you fix one of the variables, the function is continuous. It is the other variable, still the function is continuous, but jointly this function is not continuous. So, for that, let me also give you some more inputs, so that you are able to do. So, let me write 3. So, let us take a function f in a domain D in order to f continuous. Of course, continuity is continuity at a point. We are not looking at continuity of the whole domain, saying if f is continuous at a point, then both coordinate functions are also continuous at that point. Conversely, we said no, the function, both coordinate functions may be continuous at a point x0, y0, when x0, y0 are fixed independently, but jointly the function may not be continuous at that point, x0. So, let us say continuous at it x0, y0. So, let me draw a picture again here and here is the domain and here is the point x0, y0. We are saying the function is continuous at this point. So, what does continuity say? It says, so here is where it is going, so here is the value at f at x0, y0. Now, what does continuity say? Continuity says, given a neighborhood of epsilon neighborhood of the point, the value, there is a delta neighborhood of, so there is a delta neighborhood of this size that whenever I take a point inside, I take a point inside here, it value goes inside here. So, we can think of this domain, this is the, and the value goes here. So, that is continuity. If you look at the sequential thing, whenever a sequence converges to this point x0, y0, so if a sequence xn, yn is converging to x0, y0, it is going to come inside that domain and hence it will come inside this. That is the equivalence of the sequential definition and the epsilon delta. So, the point is, how does the sequence x0, y0 approach this point x0, y0? How does the sequence xn, yn approach? In the real line, you can approach a point from the left or from the right, but in R2, for example, you can approach why you could go along any line through that point or you can go along a zigzag path going through it. So, it should mean whichever way you approach the point, the corresponding values should come closer. So, another way of saying that would be, so let me just write continuity means what implies continuity at x0, y0 means, I can make it more precise means for every path in D, for every path in D going through x0, y0 the corresponding values along this path, there is so much of English, but it is very easy to understand along this path approach f of x0, y0. Let me just draw a picture of this and bigger picture, so that you understand what I am saying. So, here is the domain and here is the point x0, y0 and this is the value at the point f of x0, y0. So, continuity says suppose I approach this point x0, y0 along this path, so this is a path you can call it as G, keep on the picture. So, what I am saying is I am going to approach this point along this path, in the real line you can approach only from left and from right. So, here I can approach this point along this path, so I can take a sequence if you like going along this path to this point, so xn, yn, so this is the, so I am going to this. Then the corresponding values should come closer to the value f of x0, y0. So, let us, why this thing is useful, so let me look at an example probably to illustrate this. So, let us look at the function f of xy is defined as, let me write x plus y divided by x minus y. Now, x minus y is in the denominator, so I should remove x not equal to y. So, what is the domain of this function? Where is this function defined? It is defined at all points x, y belonging to r2, such that x is equal to, so what is x is equal to y, that is a line. So, the domain of this, so I should remove that line. So, this line is not part of the y equal to x, on this line, except on this line everything else function is defined. I want to know, so this function is not defined at 0, 0. So, f 0, 0 is not defined, it is not defined, because the line y equal to x plus is through it. Question is, can we define some value to f at 0, 0, so that the function becomes a continuous at 0, 0, so that the function becomes continuous. At 0, 0 it is not defined, function, if I want to say function is defined at 0, 0 also and continuous, so what should I do? I should look at the limit of the function at that point and if the limit exists, that should be the value of the function at that point. So, the question is, does this function have a limit? So, the question limit x, y going to 0, 0, f x, y exists or not. So, what is f of x, y? So, that is x minus y, x plus y divided by x minus y, x not equal to y. I am looking at limit of this at 0, 0. Now, if the limit at this point is to exist at 0, 0, then what should happen? Then whatever path I take going to 0, 0, if I take a sequence x n, y n, x n, y n, x n, y n, x n, x n, y n, going to 0, 0, then f of that should have some value, whichever the path may be along this path or this path, or on this path or whichever way I want to go. So, let us try to test it for some nice paths. Let us take a path like this, which is a line through the origin. So, let us approach the point 0, 0 along a line other than y equal to x, because y equal to x, it is not defined. So, let us take a line. So, consider f x, y along, so what is the equation of some other line? y equal to m of x, x not equal to 0, of course, 0 is not defined. Then what is the value of the function? f of x, along this line, what is the function? y is m of x. So, x plus m of x divided by x minus m of x, and what is that? That is 1 plus m divided by 1 minus m. So, let us assume m is not equal to 1. So, I am taking the line where m is not equal to 1, of course. This is the line y equal to m of x anyway. So, along this line, the function always is a constant. So, limit will exist. So, along every line, the limit exists. So, let us write that first, what is the observation? So, along y equal to m of x, limit x going to 0, f of x, m of x is equal to 1 plus m divided by 1 minus m, where is the constant function. Now, if the limit at 0, 0 has to exist, that limit should be independent of the path. Left limit in the real line should be same as the right limit. So, here it depends on m. So, this limit exists along every path, every line passing through 0, 0, but limit is different. So, limit at 0, 0 does not exist, but this depends on m. Hence, limit x, y going to 0, 0, f of x, y does not exist. So, for this function, the limit along 0, 0. So, I cannot define this function in any way to make it continuous. So, the important thing is the path in R 2, R 3, R n, there are more than one. In real line, only left and right possible things are there, but in R 2, R n, infinite number of ways. So, it should not depend upon. So, how is this useful? To prove that the function is not continuous, if we can show it along two different paths, the values are different. Then the function will not be continuous. So, that is continuity in R 2 or similarly in R 3. So, coordinate function continuous implies. So, what are coordinate functions? When you are fixing one coordinate, you are going vertically or horizontally. That is the path you are going. When you are fixing, say y naught, say y naught is fixed now. So, you are going horizontally, y coordinate is fixed. So, you are moving horizontally, x coordinate vertically only, but not only that, it should be along every path. So, no wonder that coordinate functions continuous need not imply continuity of the function jointly of the two variables. So, many times we tend to push the problem on one variable in the sense that to check a function is continuous in two variables. If you can show it is not continuous in one variable fixed, then it is not continuous jointly also. So, these are necessary conditions. Continuity in each variable is necessary for a function to be continuous, but not sufficient. But necessary things are always useful. Proving something is not that. So, let us, I think only two minutes left. So, let me stop that here. So, next time we will start looking at differentiability of functions.