 So, now, let us start looking at functions defined on real line. So, we are going to look at now, next looking at functions f will be defined in some domain subset of say R n to R. Let us look at what I want to do to motivate that. Let us look at the following situation. Suppose C is a point belonging to R n such that let me write every ball, every open ball at C intersects. So, this is an assumption I am making. So, what does it mean? What I am saying is C is a point. So, I think let me mention C may belong to D or may not. f is a function with a domain D. Look at a point C, that point C may or may not be in the domain, function may or may not be defined, we do not bother about it whether it is defined or not. But the domain should have the property, there are points close to C in the domain where the function is defined. So, one way of ensuring that is every neighborhood of the point C intersects D. For example, at C I can take a ball of radius 1 over n small or any epsilon that should intersect D. So, points close to C are in the domain, that is assumption I want to make. The function is defined at points as close to C as you want, but we do not know whether function is defined at the point C or not. So, that is the situation I am having. The question is, by looking at the values of the function at points close to C where it is defined, can I predict some suitable value for the function at the point C? Is the question clear to everybody? Yes or not clear? So, let me repeat again, I want to look at the problem F is a function in some domain, I am given and C is a point which may or may not be in the domain, it may or may not be defined, we forget whether it is defined or not. But positively the function is defined at points close to C. How close? As close as you want, function is defined. So, I know the behavior of the function at points close to C, that is the idea. I want to predict what could be the behavior of the function at the point C by looking at its behavior at points nearby. So, that is a prediction problem I am looking at. I am trying to predict a suitable value for the function at the point C by looking at its values at points close to it. So, what could be a way of doing that? One way could be that, let us try to come closer and closer to C, approach C and close to C the function is defined. So, look at the value of the function at points close to it, look at those images. If they are coming closer to something, then that could be a suitable value for the function to be predicted. Is it a natural thing to do? Is it ok? Intuitively very clear. So, how do I approach a point in D? I can take a sequence in D which is converging to C. Then the points of the sequence will be coming close to C as close as you want. Is it ok? So, that is why I put the condition that close to see every neighborhood it must intersect. So, there are sequences which are coming to C. So, let us look at a sequence one way is, so I have not written what is the problem. So, let me write the problem. So, the problem I am looking at is to predict a suitable value of F at C by looking at values of F at points close to C. So, that is what we are trying to do. So, one way could be, so one way is let A n belong to D and A n come to C as look at, I am trying to predict a value of the function at the point C. At a point close A n, this is the value. So, look at A n, ask and analyze, does the F of A n come closer to a value? What is the meaning of that? Converges to some value alpha. So, what we are saying is look at sequences in the domain which converts to C and look at the image sequence for those points F of A n. If they all converge to some value, then that value could be a suitable value. As I am approaching the point C, I am approaching the value alpha. But it should happen one sequence A n is converging to the C and B n is also converging. But F of A n actually converges to some value and F of B n converges to some other value. Then we will be in a problem which value is a suitable value. So, we should have for every sequence A n converging to the point C, F of A n should converge to the same value. Then that value will be a suitable value for the function at that point C. So, we are trying to predict by looking at the behavior at nearby points. We are not concerned what is the value at the point C. Is it okay? Yes, clear the problem. And if that has happened, one says the function has a limit at the point C. So, let us put a definition. We say F has limit at say X is equal to C. If for every sequence A n in the domain A n converging to C, F of A n converges to alpha, we write limit X going to C, F of X is equal to alpha. So, keep in mind limit is something to which the function is approaching as you approach the point in the domain. So, naturally conditions are there should be a sequence at least one sequence converging to that point in the domain. Otherwise, what are you going to predict? You are not going to predict anything, right? So, close to C there should be points in the domain as close as you want so that you can analyze the behavior of the function at points close to C. And once that data is given to you, you can analyze the behavior of the function at points close to C. And once that data is given, I look at the behavior of the image sequences. If all the image sequences converge to the same value, then we say that is the value the function should take. And we say mathematically that function has a limit as X goes to C. Clearly function need not be defined at that point. It is a suitable value for the function by looking at its behavior nearby. That is the way we should understand the limit of a sequence, right? And that is the way we have understood. For example, for a sequence, what was the limit? For a sequence a n, the limit a n need not be equal to C at all, right? Only they are coming closer and closer, right? That was the thing. Sometimes they may be equal to the value that does not matter, right? But we do not bother about whether elements of the sequence are taking that value of where it is going to converge or not. Same for the function, the limit is something which we want to predict a suitable value, where the function is coming closer and closer to some value. As in the domain, you come closer to the value C. And that closeness mathematically is measured by a sequence in the domain converging to C. That is, a n is converging to C. So, a ns are coming close to C. f of a ns are coming close to some value alpha, whatever be, right? So, if you want to look at a picture in R n, say R 2, so this is the domain and here is the value alpha. So, for any point, for example, the point C could be here. So, if you take a sequence X n which is converging to C, so look at f of X n, right? It could be here, it could be here, but they are all coming closer to the value alpha. And you see, in R n, there could be many ways of approaching C, there could be many paths, right? But whatever way, whichever way X n is converging, we know how to X n converge absolute value of or the magnitude of X n minus C goes to 0. We know that, the distance goes to 0, right? So, the path is not important, it is the closeness which is important, right? And in case the function is coming closer to 0, it is defined at the point C. Supposing it so happens that the point C is in the domain, that means the function is given some value. That may or may not be same as the value that we are trying to predict, right? So, if the value we are predicting for the function at the point C by looking at its behavior at nearer points happens to be same as the value of the function at that point C, then it is natural to say there is a continuity in the behavior of the function. Whatever we are trying to predict, function behaves nicely, there is a continuity in this behavior. So, that gives us a notion of what is called continuity of a function at a point. If the value we are predicting is the actual value taken by the function. So, continuity of the function at a point means the point is in the domain, right? That means f of C is defined and for every sequence xn converging to C, the limit is equal to f of, right? So, value predicted is the actual value taken. That is how you should understand what is called the continuity of a function at a point. All of you have done these limits and continuity in your previous courses, but just this perspective you should keep in mind. So, let me define. So, limit we have defined. So, let us define what is called definition f defined in a domain in Rn to R, C belongs to D, f is continuous at x is equal to C. If limit C going to x going to point is C, f of x is equal to exist and is the value of the function at that point, f of C. The value predicted is the actual value taken. So, that is continuity. Now, you all must have done this kind of theorems. Namely, if the limit, you see now what you are looking at? We are looking at some property of functions now, right? So, our domain of attraction or investigation is the class of functions defined on real line or Rn. Now, you can add functions. So, what are the operations possible on this class? You can add functions, you can multiply functions, you can divide functions, you can compose functions, right? And the theorems are about the limits, which you have already done. If f has a limit at a point C, g has a limit at a point C, then f plus g has a limit at the point C and the limit is equal to some of the limits, right? Similarly, difference, product, you can do also quotient. You have to keep in mind that the limit of the quotient is not equal to 0, as in sequences, we have done that. So, those are called algebra of limits. Limit of the sum is equal to some of the limits, limit of the product is equal to product of limits and so on, right? I am just recalling you what you might have already done. So, translate this for continuous functions. If f and g are continuous, f plus g is continuous because the limit is equal to some of the limits, right? Product of continuous functions is continuous. Composition of composite functions is continuous, yes? Let me just indicate. So, supposing f is a function and say that g is another function that f plus f composite g is defined on some domain, right? I want to say that is continuous at x is equal to c. So, f composite g at c is defined. So, that is assumption, right? For composite of functions, it is not always that f composite g is defined whenever the range of g is in the domain of f. It will be defined, right? To show continuity, let x n converge to c. We want to analyze f composite g of x n, right? What is that equal to by definition? It is f of g of x n, right? And where is x n converging to c? But I should have now g of x n converging somewhere. Otherwise, I cannot say anything. So, condition is if f composite g is defined and g is continuous at c, where domain and f is continuous at the g of x n. Then, this is continuous and this will converge to f composite g of c. So, appropriately you can put conditions f composite g, right? So, what should happen? I am looking at the point c, limit at the point c. So, first of all f composite g should be defined, right? First g is operating and then f is operating. If x n is converging to c, look at the image that is g of x n that should converge to g of c, right? So, g of x n is converging to g of c. f is defined there and if f is continuous at the point g of c, then f composite g n will converge to f composite g at the point c. So, those are the appropriate conditions that you should put by looking at what is to be done. So, conditions are not put just for the sake of being happy is because each one is required at some stage, right? Is that clear to everybody what I am saying? All of you have done this, but I am just revising and trying to make you understand how you should understand something, right? So, we will not spend time on composition of on these properties of continuous functions and so on. What we are going to look at is some properties of continuous functions. So, let me look at algebra of limits, algebra of continuous functions and we are assuming that you have all gone through and if you are not gone through or you would like to be more comfortable, do it again, right? Look at the proofs of limit of the sum is equal to sum of the limits and so on, ok? Be happy so that you understand. Be, I think the other way around, understand and be happy. You will be always be happy if you understand, right? So, properties. So, what we are going to look at is we are going to look at properties of continuous functions vis-a-vis those special properties of subsets of real line. For example, if the function is defined on a interval, domain is a interval, what can you say about the range of the function? Can I say it is the interval? If the domain is the interval, can I say the range is the interval or another property? If the domain is a close bounded interval, can I say the range should be a close bounded interval? So, such kind of properties we want to look at of continuous functions. So, continuous functions vis-a-vis special properties of subsets of the domain. So, the first one is let f be a function defined on i to r, where f is continuous, ok? Why write where b? So, let f be continuous, continuous such that, be continuous, ok? Let me just write, there is no such that meaning. If i is an interval implies f of i is an interval, I want to look at this property. So, let us see how does the proof work? I want to look at. So, i interval f of i is an interval. That is the question we are looking at. How do I prove f of i is an interval? How can I prove? The only way we know is, take two points in that set, take a point in between. That should also be in that set. So, let us take alpha belongs to f of i, beta belongs to f of i and either alpha will be less than beta or right? So, let us write alpha less than beta. Let alpha less than gamma less than beta. Take a point in between, right? We have taken two points in f of i and we are looking at a point in between. Claim, we want to analyze, gamma belongs to f of i, right? Now, what is the meaning of saying alpha belongs to f of i? Means what? Trying to understand. That means alpha is in the range of the function. You see what we are trying to analyze the range and we know only something about the domain that is interval. So, somehow I have to shift my attention to the domain and use the fact that domain is an interval. So, the proof has to be in such a way, I have to drive my proof in such a way that it goes to the domain somehow and then use that fact. I can do something there only. So, how do I go to the domain? It belongs implies there is some x belonging to i such that f of x is equal to alpha. Beta belongs to f of i. That is same as saying there exists a point. So, let us call it x1 and call it f of x2. There is a point x2 where the value beta is equal to beta. At a point x1 because it is alpha is in the range, so some value, some point x1 should be mapped to alpha. Another point x2 should be mapped to beta. Now, x1 and x2 both are in i. So, now the idea is, so let me draw a picture probably that will be, I am going to give you a picture which is only sort of real line. So, here is the domain, here is the function, here is x1, here is x2, here is the value alpha and here is the value beta and here is the value and gamma is in between. So, here is the point. Now, I can directly work with alpha, beta, gamma and so on. Straightly proof will be understandable more, it does not matter. So, let me, I think probably do it straight away here itself. Let me write, so draw this line. So, what I want to show? A means to show that gamma belongs to f of i. That means what? There is a point in the domain where the value gamma is taken. There is a, is it okay? So, I want, so we want to show that there is, this is the value gamma and what is, geometrically if you look at, what is the meaning of saying that the function will take the value gamma somewhere. I am intuitive notion of a function, if you will have a picture of the function, what does it mean? What is the picture of a function? You have all done calculus. What is the picture of a function? There is a graph of the function. Graph is the picture of the function and if you want to say that the value gamma is taken somewhere by the function, that means the graph must intersect that line somewhere. That is a geometric way of saying. So, you want somewhere, the graph, the graph should cut somewhere at a point. That is the point we are looking at. That is the point we are looking at. So, how do I capture that point? That is a question. I want to capture this point in the domain. So, how do I capture it? That is a question. So, now you see, one way of capturing this point is, at this point the value is alpha, x1. So, what I can do is, I can start looking at points bigger than x1, where the value is, this value alpha is less than gamma. So, look at all the points where the value is less than gamma and try to look at the largest of these values in the domain. See, I am going to look at the domain only. Is it okay? So, look at the largest. So, possibly that value, say at this point, the value still remains below gamma, below the line. At the point, next also it remains below gamma. So, look at the largest possible value in the domain bigger than x1, where the graph stays below that line. So, if I have the largest value, that means, after that it will, it has to cross over. Otherwise, it will not be the largest. So, that is one way of analyzing the proof. So, what should we do? So, the proof starts by looking at or, so let us look at the set a, all x belonging to this x1, x2, such that f at x is less than gamma. Is it okay? Because, so note x1 belongs to A, right? x1 belongs to A. Is that okay? Because the value at x1 is alpha. So, A is not empty. I am trying to look at the largest value. I have to ensure the largest value exists. So, how to ensure the largest value exists? The only way I know is LUB property of real line, right? So, x1 belongs to A implying A is non-empty, right? A is bounded above, yes. Why it is bounded above? Because it is inside x1, x2, right? We are choosing points between x1 and x2 only. So, it is bounded above. So, implies gamma equal to LUB, not gamma something else. I should write because gamma I have already used. So, let us write c equal to LUB of A exists, right? Now, where is c? Note, what is the value at the point? So, I have taken and I have gotten a point c, which is LUB of these points where the value is less than gamma. My aim is to show the value at c is equal to gamma, right? The value at the point c is equal to gamma. So, first of all, look at note c is LUB of something. So, I am saying this implies f of c is less than or equal to gamma. Why is that? c is LUB of S at A. In the LUB of S at A, there must be a sequence in A converging to LUB, right? There must be a sequence converging in LUB, right? So, I am not writing the proof. I am just trying to understand what I am saying. c is LUB. So, that means what? That means there must be a sequence in A converging to LUB. We have already seen that, right? c minus 1 over n cannot be, there must be a point, all that we have done. So, there is a sequence in A converging to c, right? So, there is a sequence. So, what is the value at that point of the sequence? They are in A. So, f of c is f of that element of the sequence is less than or equal to less than gamma. f is continuous. So, image must have the property. So, let me write here. There exists a sequence A n belonging to A, A n converging to c implies f of A n converging to f of c, because of continuity, right? f of A n, because it belongs to A, is less than or equal to gamma implies f of c is less than or equal to gamma. That is the proof. So, we have used continuity here. So, now what is the possibility? So, hence, so what we are saying is that point c is bigger than or equal to x 1 is less than or equal to x 2, right? It is less than or equal to, because x 2 is a upper bound and this is a l u b. So, it has to be less than or equal to x 2. Now, the question is, can c be equal to x 2? What is the value? If c is equal to x 2, what is the value at x 2? That is beta, which is bigger than gamma. That cannot happen, because we just now said f of c is less than or equal to gamma. So, this c cannot be equal to x 2, right? So, let us write no, because f of x 2 is equal to beta is bigger than gamma. So, x 1 less than c less than x 2, right? It has to be, but still we want to show that f of c is equal to gamma. It is less than or equal to gamma. If I can show it is also bigger than or equal to gamma, we are through, right? But this implies there is a sequence b n such that c is less than b n less than x 2 and b n converging to c. Is that okay? Because if c is less than x 2 and there is a gap in between, right? I can always have a sequence coming to c on that side. And where are these b n's? And what can say f of b n's? b n is on the right side of alpha, which is a supremum, where the value of the function is less than. So, it has to be bigger than or equal to. That is the only possibility. Alpha is the l u b of all points in x 1, x 2, where the value is strictly less than gamma. So, nothing bigger than l u b at that point, the value has to be bigger than or equal to gamma. Is that okay? Because b n bigger than c. Is that okay? No, not okay. Here, I am just, I am, look at the graph. You will understand it better. So, here is c and c is less than x 2. So, there is b 1 and so on. b n converging to c. What can be the value at this point? It has to be bigger because what is c? c is the least upper bound of all those points, where the value is less than gamma. So, something bigger than that least upper bound, the value has to be bigger than or equal to gamma. So, these are points here in this point. So, that means what? So, implies f of c by continuity again is bigger than or equal to gamma because f is continuous. Each element of the sequence b n f of b n is bigger than gamma. b n is converging to c. So, where does f of b n converge? It converges to f of c by continuity. Each b n is bigger than or equal to gamma. So, limit has to be also bigger than or equal to gamma. So, this is second and the first one was it is less than or equal to. So, this is 1 and you can call this as 2 if you like 1 and 2 imply f of c is equal to gamma. So, we are not doing anything surprising very naturally. What we are saying is here is if we look at the picture at this point the value is alpha. So, let me start travelling along this route. Go on moving till you remain below the graph and look at the largest value of this where you stay below. That must be the point where you cross over. That must be the point that you cross over. So, written mathematically look at all the points in x 1, x 2 where the value f of x is less than gamma. That is a non-empty set less than or equal to gamma. Yes, that is a non-empty set. It is bounded above by x 2. So, it must have a least upper bound and that least upper bound is the required one because everything on the right hand side will be bigger than or equal to. Here it is less than or equal to that. So, it has to be equal to gamma. So, that is one way of proving that if a function takes two values alpha and beta then it takes every value in between. Another way of saying the same result what we have shown if i is a interval f of i is a interval. But what is the meaning of saying? So, if i is a interval alpha and beta are two values then every value in between also is taken that is what we are showing. So, this theorem can also be interpreted as saying for a continuous function if two values are taken alpha and beta then every value in between also must be taken at some point. So, in that way it goes by the name of intermediate value property for continuous functions. So, the proof that image of a interval is the interval also gives you what is called intermediate value property. It is same. So, corollary if yeah will ok what is the doubt? Where is b n is on the right side? b n is on the right side of c and what is c? b n is not belong to a. Who said b n belong to a? b n is the set of those points in a where f of x is less than gamma. It is not all x 1, x 2. Look at the definition of a. What is a? x belonging to x such that f of x. So, for all points of a f of x is less than gamma and we are taking the supremum of it. Supremum of a is alpha. So, any point on the right side of alpha cannot be an element in a because alpha is the least upper bound of a. So, if it is not in a at for any point f of x is less than gamma. So, it is not. It is outside. That means what? The value should be bigger than or equal to gamma. So, for all points on the right side of c, the value is bigger than or equal to gamma. Only we have to ensure that there are points on the right side and for that c cannot be equal to x 2 because f at c is less than or equal to gamma. We have just now shown f of c is less than or equal to gamma. So, c cannot be equal to x 2 because the value at x 2 is beta which is bigger than gamma. So, there has to be a gap in between. That means c has to be strictly less than x 2. So, it is an interval in between. So, take any sequence converging and that does the job. So, I was writing another way of this or a consequence of this, whichever way you like. If f of x 1 is equal to alpha, f of x 2 is equal to beta and f is continuous in x 1. I am not saying x 1 should be less than it could be other way around. x 2 could be less than x 1. So, let us say x 1 less than x 2 or it could be other way around. Then there is a point c belonging to x 1, x 2, say that f of c and gamma is, if f is continuous for alpha less than gamma less than beta, there is a point c, say that f of c is equal to gamma. So, if a continuous function takes two values, alpha and beta, it must take every value in between. So, that is what is called intermediate value property. This is very useful result. For example, at some, you want to locate where does the graph of a function, let us first interpret this geometrically. Geometrically, this says, if you are here and you want to go here, there is a value at alpha, there is a value at beta. If you want to go and every value must be taken, this value will not be taken. If you go up to here and then start your graph somewhere else. So, geometrically saying intermediate value property holds for continuous function means the graph of the function has no breaks. That is a geometric interpretation of this. So, continuity of a function at a point means, if you start somewhere and domain is an interval and you end somewhere, then once you start drawing the graph, you should not lift your pen or pencil, whatever you are doing to draw the graph. You should continuously go on doing. There is no break in the graph of the function. This is why it is important in calculus when you want to get a picture of the function. This is the first tool which gives you a picture of the function. Namely, continuity implies there is no break in the graph of the function. Here is another way of proving this. That is interesting. So, let us just look at x1, at x2, here the value is alpha, here the value is something beta and here is gamma in between. See, we are trying to locate a point in between x1 and x2 where things cross over. So, another very intuitive way of doing this is the following. At x1, the value is alpha and at x2, the value is beta. What I can try to do is, look at the midpoint of this and look at the value at the midpoint. So, look at f of x1 plus x2 by 2. What is the value of the possibilities? Either it is equal to gamma, that is a place where actually the function crosses one possibility. You are very lucky or it is below or it is above. Only three possibilities. If it is cutting at that point the line gamma, then we are through. If not, let us assume it is below. So, let us assume the value here. It is still below. Now, value at x2 is above. So now, let us only concentrate only in this part of the graph. The original one, at x1 it was below, x2 it was above. Now, I have sunk my vision. I know, I do not forget about everything else. Look at only the midpoint and x2. At the midpoint, the value is below. x2 value is above. So, you see now I am trying to capture that point kind of a thing. Now, again I will look at the midpoint of that. Again, possibility is same. Either I hit the jackpot, I get the value or it is below or it is above. So, if it is below, I continue that process and go on shrinking. Now, if at this stage the value is above, supposing it was above, then I will concentrate only on this part. So, what I am doing is at x1, x2, x1 the value is less, x1 the value is less, x2 the value is bigger, cut it into half and keep only that half that at one end point the value is less, other end point the value is more. Again, cut it into half and see which one is working. So, go on shrinking this interval. At one end point the value is less, other end point the value is more. So, what you get? You get a nested sequence of intervals such that at the left end point the value is less, at the right end point the value is more and the intersection of all of them has to be a single point. What will be the value at the single point? It has to be equal to gamma because if I look at the left end points, it should be less than or equal to gamma. If I look at the right end points, it should be bigger. So, they have to be same. So, this is called the bisection method of capturing a point making it smaller end. So, you can write down the, I am not writing the proof, it is second proof. You can try to write out the proof yourself. Basically, the idea is x1, x2 at x1 the value is less, x2 the value is bigger. Cut it into half, either this half or this half will have the same property again. At the left end point the value will be less, at the right end point the value will be more. Go on doing it. If you do not hit the jack pot in between, then eventually you should capture that point.