 So, let us look at some examples and then it will help us to understand. So, let us look at some examples. Let us look at why we started all this was, let us take, let F be AB to R be monotone. Say monotonically increasing. Let us say it is monotonically increasing. So, let us take a partition. Let, I want to calculate the variation and check whether it is a bounded variation or not. So, let us take any general partition X n. So, F AB to R, I did not write F AB to R be. So, let B be a partition. Then, what is F of X i minus F of X i minus 1 absolute value? Function is monotonically increasing. X i is on the right side of X i minus 1. So, this value is equal to F of X i minus, no need to put absolute value, because the function is monotonically increasing. So, implies what is the variation of F with respect to the partition P? So, what will be the variation? So, we said it by definition it was sigma 1 to n F of X i minus X i minus 1 absolute value and that is equal to, there is no need to put absolute value. So, that is sigma i equal to 1 to n F of X i minus F of X i minus 1 and what is that? Now, terms cancel out now. So, it is F of B minus F of A. So, whatever B implies, V AB of F exists equal to F B minus F of A. Hence, F is of bounded variation. It is a function of bounded variation. If it was decreasing, only difference would have come here. This would have been reversed. So, then it will be F A minus F of B. So, same proof works except for difference of that sign bounded variation. Similarly, let me write F monotonically decreasing implies F of bounded. Let us look at another example. Let us look at the function just now. We looked at the indicator function. So, consider F of X is equal to 1 if X belongs to, is a rational inside AB. It is a rational number. The value is 1. It is 0 if X otherwise. So, that means if it is irrational, the value is 0 and the rational value is 1. So, what is it? It is an indicator function of the rational in AB. It is an indicator function of. But it takes only two values. So, what do you think is the variation of this? It is not monotonically decreasing, neither monotonically decreasing. So, what do you think is a variation of this function? It looks like, yes, it looks like 1. Now, how much? You see, variation is how much up and down it sort of goes. That was the right. But this function is going up and down very often. At rational points, it goes up and in rational point, it comes down. So, it looks like it is varying too much. So, it is not a function of bounded variation. So, claim, guess is F is not bounded variation. This is only a guess, because it is going up and down very frequently. So, how does one write that? That idea, I can visualize it up and down, but how do I write it? So, the idea is, here is AB. And I want to say that the variation, I can find a partition, where the variation is equal to n, say. For any n, any natural number n, I can find a function, find a partition, so that the variation on that partition is equal to n. Then, I take the supremum, it will be infinite. So, for example, if I take the point a, let us take a rational here and a irrational here. So, some point, let me x and y, where x is irrational and y is the complement of that. So, what do you want me to write? I will write AB minus, let us write, rational itself. So, at this point, this is a rational point. So, what is the value at the rational? It was 1. At this point, the value is 0. So, if I take a partition in which these two points come, then in that summation, at least number 1 will appear. So, I can choose any finite number of such pairs, n number of such pairs. So, the summation of that variation over this partition would be at least n, because there will be something contributed by this, something contributed by that also. So, it will be bigger than or equal to n. So, let me write. So, we can choose, for every n, we can choose points xn, let us write a less than x1 less than y1, points. So, let me write xn, yn such that, let me, I want to write xn, yn or xk, yk. Let me just improve it. Choose points xk, yk, where k is between 1 to n such that, a is less than y1, is less than x2, less than y2, is less than xn, less than yn, less than b. So, the pairs. So, keep in mind, I am looking at the pairs in between. So, this is one pair. So, this is one pair, this is another pair and this is the nth pair, where what do we want? We want such that, each xi or each xk is rational, each yk is irrational. Then, if I call that as the partition p, if I call those points as the partition p, then the variation a, b, f or call it as pn, if you like, pn will be what? It will be value at x1 minus the value at a, absolute value plus absolute value y1 x1 plus value at difference between the absolute value of x2 minus y1. So, I am just saying that, this is bigger than or equal to sigma f at yi minus f at xi. Other terms, I just drop, only keep the values difference. So, which is equal to n, because xi is rational, yi is irrational. Is that okay? Other terms, I will just drop. So, instead of equality, it is bigger than or equal to null. So, implies vab of f is not finite. So, one writes it as plus infinity. So, this function, very nice, very simple, is not a function of bounded variation. Many examples you can construct of such things. So, probably, I think, let me just say what is right. So, let me, because we do not want to prove too many things for the functions of bounded variation. Probably, in a higher course, you will come across these things. So, let me just, because I am not writing. So, it is, we had already analyzed many things about monotone functions. So, let us, so it has jump discontinuities. They are countably many and so on. We already had looked at continuous function, which is 118 strictly monotone. Those properties we already looked at. So, here is, so let me revise. So, saying that a function a, b to r, this is a partition. Then, this is called the variation of f with respect to the partition p. And look at the supremum, that is called, normally called the total variation. And if it is finite, we say the function is a bounded variation. So, here is, every monotone function is of bounded variation, that we just now saw, is a, b. Here is another one. You have this Lipschitz function. Remember, what was the Lipschitz function? Let me just say, what was the Lipschitz function? Another example of, so f, on any domain actually, let us write it on a, b to r. We said it was Lipschitz, the German mathematician Lipschitz, who defined it first, if there is some alpha such that f x minus f y is less than or equal to alpha times x minus y, for every x, y belonging to a, b. So, how much change in f comes with respect to the change in, is a directly proportional kind of a thing, less than or equal to alpha times change is less than. We, this we had come across actually, when we said that every Lipschitz function is uniformly continuous. So, alpha times epsilon delta or you can write sequence, whenever a sequence x and y n goes to 0, that will imply f of x and y n also goes to 0. So, uniform continuity. We are saying that this is a function of bounded variation also, because when I look at f, that variation f of x i minus f of x i minus 1, that will be less than alpha times x i minus x i minus 1, so that summation terms will cancel out, what we left is b minus a. So, the variation with respect to any partition is less than or equal to alpha times b minus a. It is something like monotone function, but even if it is not monotone, it is still true. So, every Lipschitz function is a bounded variation, because variation a to b, f with respect to b will be less than or equal to alpha times b minus a for every partition p, because of this inequality. So, it is of bounded variation. Just now, we saw that this function is not a function of bounded variation, it is not of bounded variation. So, we can give some more examples. Here is one example, this function, this also we had come across while trying to do something about connected subsets of R 2. So, as you come near 0, the graph goes up and down very fast. So, again you can see that the variation of this function will be infinite, because you can always pick up points where the value is 1 n minus 1, points in 0 1. So, this function is not a bounded variation. So, here are some properties which will not prove, but just try to understand why. You will come across these things probably in a higher course. If a function is a bounded variation, then it should be bounded function. Is it clear? Because if it is unbounded, that means you can go on increasing the values. So, you can enclose those points in the partition if you like. Or you can simply prove also mod f of x is less than or equal to mod of x minus a plus mod. So, trivial partitions you can choose. There is this function which are bounded, but not of bounded variation. Just now we example that 0 1 rational. So, it is a bounded function. It takes only two values. It is not a bounded variation. Bounded variation functions have nice properties. So, this is called the algebra of functions of bounded variation. f and g are bounded variation, then f plus g f minus g f g alpha times f. All are functions of bounded variation. Every monotone function is a bounded variation. One can define many things like you can define the variation instead of full interval a, b only in the part a to x for x in a, b and show many properties. So, just go through these properties once, but do not look at the proofs because we are not going to ask you. Basically, if c is a point in between, then the variation over a to c is less than or equal to variation over the whole interval. That is obvious. It says the variation adds up. Variation a to c plus c to b is same as variation a to b. They are functions of bounded variation. So, these are the properties. This a to x, x belong to a, b. This itself is an increasing function. Using these properties, one can prove this is also an increasing function.