 So, we were looking at connected subsets of Rn and we were trying to prove a theorem. We had given illustrations of that theorem namely, A contained in Rn is connected if and only if every function f from A to 2 points at 0, 1 continuous is constant. So, we looked at many consequences of this theorem. So, let us give a proof. Suppose A is connected and f is a function from A to 0, 1, f continues. So, it is a function, continuous function on A taking two values 0 and 1. We want to show it should be taking only one value, it should be a constant function. So, if f is not constant, then what will happen? There is at least one point where it takes the value 0 and some other point where it takes the value 1. Then, let us look at the inverse image of 0 and inverse image of 1. So, all the points which go into 0 and all other points which go into 1 are non-empty proper subsets of A. Because there is a point which goes to 0, there is another point which goes to 1. So, that means 0 has a pre-image, 1 has a pre-image. So, they are non-empty and this pre-image is of 0 and 1. They are not whole of A, because different points are going to 0 and 1. The only observation is that this set. So, let us observe f inverse of 0 and f inverse of 1 are both close subsets of A. They are both close subsets of A. Is it okay for everybody? It is both close subsets because if you take a sequence in f inverse of 0 xn converging to x, then f of xn is equal to 0 for every value. So, f of x also is equal to 0 or you can look at singleton is a closed set. So, inverse image of closed set must be closed by definition of continuity. So, both close subsets of A and A is equal to f inverse of 0 every point of f inverse of 1. Every point of A either goes to 0 or 1. So, inverse image should cover everything. So, what we are saying? So, this f inverse of 0. So, you can look at that f inverse of 0 is a subset. What is this complement in A? That is f inverse of 1. So, that is close. So, its complement is f inverse of 0. So, both are open and close subsets of A because both are f inverse 0 is close. Its complement is f inverse of 1 which is also close. So, that is both open and so and further f inverse of 0 f inverse of 1 over both open and close subsets of that is not possible contradicting this contradicts connectedness. Because if A is connected it should not have any non-empty proper subset which is both open and close. So, let us look at the converse. Conversely, let us have the property that every function every continuous function every continuous function f A to 0 1 is every continuous function is constant function. So, that is given to us to show A is connected. So, that is what we are saying. So, if not again by contradiction suppose not then there exists a non-empty subset U of A proper non-empty subset of A which is both open and close in A. So, there is a non-empty proper close non-empty proper subset U of A which is both open and close because it is not connected. So, it should have. Now, here is something which we will find useful later on also. Consider, let us look at the function f defined on A taking values in 0 1. So, how I am going to define f of x is equal to 0 if x does not belong to A and it is 1 if x belongs to A. Sorry, not A, A is the main. So, let us the set I want to write is U. We have a set U which is both open and close. It is proper. It is non-empty. So, define f of x to be equal to 0 if x belongs to U and x does not belong to U and if x belongs to U then put the value to be equal to 1. So, it is defined everywhere on A. Is this function continuous? What do you know? Is this function continuous? To check continuity what you have to check? What is f inverse of 0? Let us check any inverse image of every closed set is closed. So, what are closed subsets in the range? Either empty set or singleton 0 or singleton 1. What is inverse image of empty set? Empty set itself. There is nothing which is going into anything. So, that is closed. What is the inverse image of 0? That is precisely U, not in U. So, that is U complement. Is U complement closed? Because we have assumed U is both open and close. What is the inverse image of 1? That is U which is again closed. What is left is the whole 0 and 1 together. What is the inverse image of 0 and 1 together of the range that is the domain? The whole space is always closed. So, inverse image of closed sets are closed. So, this function is continuous because let us write inverse image is of. But it is not constant. But this is not constant. But f is not constant. It is not constant function because on U it takes the value 1, on U complement takes the value 0 and the proper subsets. So, there are points in U and there are points in U complement. So, this is a contradiction. Because what was our assumption? Our assumption was whenever we have a function which is continuous in 0, 1 it should be constant. So, we have produced a function which is not constant, but it is continuous. So, that is a contradiction. So, that proves the theorem which we saw has lot of implications in giving examples of connected sets in R n. This is a very typical function. So, let me probably put a remark or a note. Let x be any set and let us say y is a subset of x. So, if I define a function f like we defined earlier on x in 0, 1 with values f of x is equal to 1 if x belongs to y and it is 0 if x does not belong to y. This is a kind of example that we had in our proof for the set U. So, what is this function doing? On any set x, this is any set x. Look at a subset y of it. So, look at a subset y. So, this is the function defined on the whole space x. So, this is y complement. So, what does it do? On y, it gives the value. The value is 1 and on y complement the value is 0. So, it is a kind of thing. If the point lies in the set y, a light goes up. If it is in y complement, light does not go up. On, off, true, false, it takes only two values. So, you will find this kind of thing coming in probability theory. So, this function indicates when the point is in the set A. When the point is in A, it takes the value 1, non-zero value. So, this is called, this function is called, this is called the indicator function of the set, of the set y. So, this is called the indicator function of the set y. Denoted by, this is a Greek letter called chi. So, this is chi, chi lower A indicating it is the indicator function of A. So, it is defined on the whole space, sorry, we are taken, not A, we have taken it as y. Defined on the whole space, taking values in 0, 1. The indicator function of y at a point x is equal to 1, if x belongs to A and is 0, if x does not, I am writing A again and again, y and belongs to, does not belong to y. On the compliment, it is 0. On the set, it is, this is called the indicator function of, the simplest kind of function. Actually, this function has a beautiful history that, historically, when the notion of function was not defined properly, there was a mathematician called Drishley, who gave this example to indicate that, functions can be very simple, taking only two values. But, they need not be given by a formula. You cannot have a formula for this f. So, till Drishley gave that example, everybody believed that, a function should have a formula or a graph. So, this was the beginning of abstract notion of a function. And here are some properties, which if you have not come across, you should. So, what is, we have got y contained in x. What is the indicator function of x? Compliment of x is 0, I am sorry, empty set. So, this is constant function 1. Let us look at two sets, A and B, subsets of x. What is A intersection B? So, you can, I leave these as very simple exercises. So, you try to prove it. That is, this chi of A multiplied with chi of B. And third, chi of A union B, this square kind of union means, A and B are subsets of x and A intersection B is empty. They are disjoint sets. Then, this is chi of A plus chi of B. Very nice properties of such a simple function. For example, you can also ask what is, in general, when they are not disjoint, this is chi of A plus chi of B. It will be counting intersection twice. So, minus indicator function of A intersection B. So, you can think of already a probability theory coming into picture kind of, I think. What more? I think this is nice. What is chi of, does everybody know what is the set A delta B? So, this is set A delta B is called the symmetric difference. This is A minus B union B minus A. So, this set is called the symmetric difference of A and B. From A, this is same as A union B minus A intersection B. So, pictorially, if this is A, this is B, then this is A intersection B. And what is the symmetric difference? The symmetric difference is precisely, so this is A minus B and this is B minus A. So, this is A minus B and this is B minus A and this part is A intersection B. So, symmetric difference. So, what is this equal to? One can show it is the absolute value of chi of A minus chi of B. So, these are very useful properties of the indicator function. These are functions, nothing, no continuity, nothing involved. So, we looked at connected subsets of R n. We looked at examples of that. So, we have looked at various properties of compact sets, connected sets, both in R n and R n. We also looked at one nice class of functions. I should state some of the properties of that. We looked at a function, monotone function. Some properties probably I should state more. So, a function f defined on D contained in R is monotone. Remember, if x1, x2 belonging to D, x1 less than or if x2 implies f of x1 is less than f of x2. If x1 is strictly less than x2 implies f of x1 is strictly less than f of x2, that was strictly increasing. So, strictly monotone, this is strictly increasing. So, monotonically increasing, let me write arrow up saying, monotonically increasing less than or equal to strictly when the inequality is strict. And so on. So, we looked at various properties. We showed that f monotone implies f is continuous or a better way of writing that would be the number of discontinuities. f has at most countable number of discontinuities that we showed. Every monotone function is discontinuous. The only possible discontinuities are the jump discontinuities and they are at the most countable. Here is a interesting thing, which is a very deep theorem. Probably I will say it when we come to differentiability. Every monotone function in fact is differentiable where? So, there is notion of what is called sets of length 0. So, probably I will state it later on when we come to differentiability. I will come to it later. Let us not go into this now. But that is a very interesting theorem and a deep theorem which analyzes differentiability of monotone functions also. So, what I want to say is the following. So, let us look at a monotone function f monotone. What kind of a is a monotone increasing or decreasing? Let us try to draw a picture of it. So, this is somewhere a and this is b. Say it is a monotone function. So, it starts somewhere. This is the value at the point a. Say it is monotonically increasing. So, probably it goes up like this and then there is probably there is a discontinuity. It goes like this and maybe it remains constant somewhere and then again starts going up like this. So, this is the value at the point b. So, this is f of a and this is f of b. Now, what we want to analyze is how much it is a monotone function? How much it can vary? How much the values of the function fluctuate in the interval a, b? Somewhere a value goes up, somewhere a value goes up. How much is the fluctuation? We want to analyze that. So, that kind of thing is measured by something called the variation. So, let us write definition and then we will come back to this kind of a function, monotone. So, f is a function defined on an interval a, b to r. So, this is the interval a, b. Let us look at what is called a partition of the interval a, b. So, let p, so a equal to x0 less than x1 less than xn equal to b. So, there are finite number of points in the interval a, b. Such a finite number of points is called a partition of a, b. You are cutting up the interval into parts, b, a partition. So, a is equal to x0, x1, x2, x3, xn minus 1 and xn. So, this is a partition. So, what we want to do is, we want to look at the value of the function. Say, in general, it will be xi minus 1 and xi. So, this is the value of the function at xi and value of the function at xi minus 1. What is the value at these two points? We do not know which is bigger, which is smaller. But, if you want to measure how much is the change in the value, let us look at the absolute value of the difference. So, that is the change in the value as you go from xi minus 1 to xi. It may be up or down. We do not know. Look at total of this. We have taken it at a general point. So, the value at a minus the value at this, this and this probably and this and this, this and this and so on. So, look at these differences. So, this is how much the function varies at these end points. This is a variation of the function at these end points. So, this is given a name. So, this we call it as v, variation from a to b of the function f with respect to the partition p. So, this is called variation of f, is called the variation of f on a, b that is a domain with respect to the partition p of a, b. So, it depends on the partition. Different partition, it will be different. But, let us observe, this is always a non-negative quantity because there is some of absolute values. So, note v a, b, f, b. So, this is bigger than or equal to 0. It is a non-negative quantity. So, if I look at the variation of f over p, p a partition, if I look at these numbers, this is a subset of the real line. All are real numbers which are actually non-negative. But, we do not know whether it is bounded above or not. It may be bounded above. It may not be. If it is bounded above, it will have a least upper bound. So, let us write, so call it supremum. Let us say the supremum of this. Let us write v a, b of f to be the supremum of this. So, what are the possibilities? This number, which is a supremum of these non-negative numbers, may be a real number. But, this set may not be bounded above. If it is not bounded above, we say the variation is infinite. So, if v a, b, f is meaning what? This set is bounded above and the supremum exists as a number. We say f has bounded variation. So, look at the variation of the function with respect to all partitions. If that is finite, supremum of that is finite, we say the function as finite variation or function is of bounded variation either way.