 So we shall continue with the discussion of limits of functions. Let us quickly recall what we did in the last class. We started with the two metric spaces x d and let us say y rho those were two metric spaces and we took a subset E of x, E or I do not know it may be A also subset A of x not empty subset and P as a limit point of A l some point in y and we define what is meant by saying that limit of f x as x tends to P is equal to l. We give the definition in terms of epsilon, delta, etcetera. I will not repeat that definition but we also proved that saying that the limit exists is equivalent to saying that given any sequence x n in A converging to P the sequence f x n in y should converge to l and this is equivalent to saying that a limit of f x as x tends to P is equal to l. And one of the major use of this is to show when the limit does not exist that is suppose you are able to find two sequences x n and y n both converging to P and suppose a limit of f x n and limit of f y n is different then obviously this limit cannot exist. We also seen some examples of that kind. Let us see a couple of more examples. Let us for example this is one of the famous example f x is equal to sin 1 by x and suppose we want to discuss whether of course this is not defined at x equal to 0 but it is defined for every x not equal to 0. It is defined for every real number x which is not 0 and since it is defined for every non 0 x if you take this set A as r minus 0 suppose it is set A as r minus 0 then 0 is a limit point of this set. So, we can talk of limit of f x as x tends to 0. So, we can talk of limit of f x as x tends to 0 but we can talk does not mean that the limit exists. So, what we can do is that for example to show that the limit does not exist one can look at the sequences for example let us let us just take a sequence. Suppose I take x n as so let us say 1 by n pi suppose I take x n as 1 by n pi then what is f of x n this tends to 0 right x n as 1 by n pi that tends to 0. What is f of x n it is sin n pi right f of x z is sin n pi and what is that for every n it is 0. So, it is a constant sequence 0. So, this tends to 0 that is fine now let us take something else suppose I take y n y n is equal to instead of n pi let me take something else suppose 1 by let us say 2 n plus 1 pi by 2 let us. So, that means pi by 2 pi by 2 things like that right this also tends to 0 this also tends to 0 and what about f of y n f of y n it is sin 2 n plus 1 pi by 2. So, that is 1 that is 1 for all. So, f of y n is 1 for all n. So, this tends to 1 this is 1. So, we have found 2 sequences x n and y n both converging to 0 f x n converges to 0 f y n converges to 1. So, limit cannot exist as x goes to 0. So, this is a simple way of showing that the limit does not exist. When the limit exists then of course first of all you have to have some idea of what that limit should be if you have some idea of what that limit should be and then one way is that you can just use this epsilon delta definition and try to show that that is that requirement is satisfied. Let us see one example of that type also. Let me just make a small modification here suppose I take this g x as this is also one of the famous function. I will multiply that by x sin 1 by x of course for x not equal to 0 and we will take the same set A. Now, let us ask the same question limit of g x as x tends to 0. So, as far as this is concerned we have shown that this does not exist and that is the argument. Now, when you want to show that the limit exists first of all you have to have some idea of what this l should be then only you can proceed. In this case what do you expect? We can expect it if at all the limit exists it has to be 0. So, we should try to show that the limit is 0. Now, how does one show that the limit is 0? Let us say we just want to use the definition. So, we shall just take let us say we are given epsilon bigger than 0 and then we want to show that for this epsilon we should want to find a delta that is we want to find delta to find delta bigger than 0 such that what should happen? Whenever mod of x minus 0 is less than delta whenever mod of x minus 0 is less than delta this should imply that mod of g x minus 0 should be less than epsilon. Let us just take this last thing mod of g x minus 0 that is same as mod g x mod g x minus 0 this is the thing, but mod x mod x sin 1 by x and we have already know that mod modulus of a b is same as modulus of a into modulus b. So, that is same as mod x into mod of sin 1 by x and now there is one obvious observation here that is this number sin 1 by x mod sin 1 by x is always going to be less than or equal to whatever be x. So, this is always less than or equal to mod x. So, what we want is that this mod x should be less than epsilon whenever mod x minus 0 is less than delta. Now the choice of delta is obvious we can just take delta is equal to epsilon we can just take delta is equal to. So, whenever mod x is less than delta mod g x minus whenever mod x minus 0 is less than delta mod g x minus 0 is less than epsilon. So, that means just take epsilon is equal to delta sorry just take delta is equal to epsilon and this works. Now, before proceeding further let me also take one more very famous example it is called it is sometimes or some books call it Lebesgue function you can define on it on any subset of r, but let us say we define your full r f from r to r will define it f of x. Let me use something else let us say suppose I call it h. So, h of x is equal to 1 if x is rational if x is rational and 0 otherwise 0 otherwise. Now, if you take in this case we will be able to show that if you take any x or any p for that matter the limit of h x as x goes to p does not exist. And how does that follow let us take for the for the time let us look at just a p is equal to 0 say limit of h x as x tends to 0. Now, as I said we want to say limit does not exist we use our strategy that is find two sequences both converging to 0 both converging to 0, but h of x n and h of y n should go to different limits. So, that is easy. So, suppose I take let us say x n is equal to 1 by n x n is equal to 1 by n that tends to 0 what about h of x n h of x n is 1 for every n h of x n is 1. So, h of x n is 1. So, that tends to 1 let me take another sequence y n suppose I take y n as root 2 by n you can take any such sequence basically it should be an irrational number y. So, y n as root 2 by n and. So, this tends to 0 this tends to 0 and what about f of y n f of y n has to be 0 f of y n has to be 0. So, this tends to 0. So, again we have two sequences x n and y n both converging to 0, but h x n and sorry this is h y n h x n and h y n go to different limits. So, the limit does not exist and is it clear to you that there is nothing in particular about 0 here. If I take any other number still one can find one sequence of rational numbers converging to it and another sequence of rational numbers converging to it and you can produce a sequence like this. So, this limit function limit of h x does not exist as x goes to p for any p limit of h x as x x goes to p does not exist for any value of p. So, let me just write limit x does not exist does not exist for whatever be p for every p in R. Now, let us see a few special types of functions and certain theorems about those functions and we will see again that we can we shall use the corresponding theorems about the sequences and we shall immediately get the conclusions for the theorems about the limits of functions. So, to do that let me we shall take this space y as R that is let take the space y as R and let us say all other things are same I am taking a as a subset of x p as a limit point and suppose we consider two functions f and g both from a to R f and g both from a to R and suppose limit of f x as x tends to p is l and suppose limit of f x as x tends to p is l and limit of g x as x tends to p is let us say some value m. Of course, remember l and m now are real numbers f x and g x those are real valued functions their domain may not be R they can come from any matrix space but the values are in R. Once the values are in R we can talk of what is meant by f plus g f into g f by g and things like that and what we want to say is that if the limit of f x is l and limit of g x is m the limit of f plus g as x goes to that is that is l plus m etcetera. So, that is the theory then first thing by the way this whole symbol means when I say limit of f x as x tends to p is equal to l it means that limit exists and it is equal to l similarly here. So, otherwise we do not write this equal to anything. So, what I want to say that limit of f x plus g x or if f plus g x you can say whatever limit of f x plus g x as x tends to p this is equal to l plus m second thing is limit of f x into g x as x tends to p is l into m and finally, if this m is not 0 if this m is not 0 then we can talk of limit of f x by g x as x tends to p and that should be l divided by m. If m is not 0 limit of f x by g x as x tends to p is l by m again I will say that we shall not spend any time in proving this just use the corresponding theorems about the sequences. Let us just see what is the meaning of this we have said that this means that whenever you take any sequence of elements in a let us say x n whenever x n in a converges to the point p f x n should converge to l and g x n should converge to m then f of x n plus g of x n should converge to l plus m that is what about the sequence we know that the sequence f x n converges to l and sequence g x n converges to l and about the sequences we already proved that the limit of the sum sequence is same as the sum of the limits. So, f of x n plus g of x n should converge to l plus m similarly f x n into g x n should converge to l into m. So, no new concept is involved and similarly if m is not equal to 0 then f x n divided by g x n should converge to l by m. Now, you may wonder that not only m not equal to 0 in order to talk about f x by g x this g x also should be different from 0, but we need not say that specifically because see the what we are bothered about is only about the values of x near the point p and if m is not equal to 0 and if g x tends to g x n tends to m then for large values of n g x n will be different from 0. So, f x n by g x n or f x by g x will be defined well defined for x close to p if m is different from 0. So, as I said all this theorem follows by corresponding theorem about the sequences and our equivalent theorem about the limit of a function and the limit of a sequence. Now, let us go to the next concept which in certain way depends on the limit and it is that of continuity. Again here also we shall take these two matrix spaces as it is x let us say x d and y rho matrix spaces and again we will take a non empty set a in x and f is a function from a to y f is a function from a to y and this time we will take this point p not a not as a limit point of a, but p is a point of a p belongs to a that means the function must be defined at that point a and then we shall define what is meant by saying that f is continuous at p. So, f is said to be continuous at p continuous at p in a if the definition is again very similar to the corresponding definition of the limits if for every epsilon bigger than 0 there exists delta bigger than 0 such that for every x in a if distance between x and p is less than delta distance between f x and f p should be less than epsilon. So, distance between x and p less than delta distance between x and p less than delta this implies distance between this in since the f x and f p those are going to be in y there the distance is rho. So, rho f x f p is less than epsilon if f is continuous at every point in a we said that f is continuous on a or in a. So, if we said to be continuous on a continuous on a or some books also use in a that is a minor point if f is continuous at every point in a you can realize that this definition of continuity and the definition of limit they are very closely related to each other. But there are some differences one difference is that while talking about the limit this point p need not belong to a the point p need not belong to a it has to be if it is enough it is just a limit point. Whereas for talking about the continuity the point p must be a point of a point p must be a point of a and since it is a point of a we can talk of what is f of p in case of limit there is no such thing as f of p f of p may not be defined at all that is the first thing. Secondly once if p belongs to a p may or may not be limit point p may or may not be limit point. But suppose p is a limit point suppose p is a limit point then you can say that this is same as saying that limit of f x as x tends to p it is same as f p because after all this is the definition of limit in this if I take it as l it will be in that limit of f x as x tends to p is equal to l. So, if p is a limit point then saying that the function is continuous at p it is same as saying that limit of f x as x tends to p is equal to f p. If p is not limit point then what suppose p belongs to a but p is not a limit point remember we had called such a point as isolated point p is not a limit point means what there exist some open wall containing p which contains no other point of a only that point p. But in that case we can say that in that case we can say that whatever epsilon is given suppose p is an isolated point essentially what I want to say is that if p is an isolated point then the function is always continuous at that point. So, let us just make that observation if p is an isolated point of a p is an isolated point of a then f is continuous at p. Now, how does this follow let us just see since if p is isolated point what should happen is that there should exist a wall with center at p which contains no other point of a let us. So, we can say that since p is isolated there exist suppose I call radius of that wall as delta suppose I take the radius of that wall as delta there exist delta bigger than 0 such that open wall with center at p and radius delta its intersection with a its intersection with a must be singleton p because it contains no other point of a it can it contains no other point of a intersection a must be singleton p. Now, is it clear to you from this that if I see what this says is that for every epsilon there should exist some delta such that whenever you take any point x in a and if I if the distance between x and p is less than delta then distance between f x and f p should be less than epsilon, but if I take this delta for example, if I take this delta then only x in a which will satisfy this is p and in which case f x and f p distance between f x and f p will be 0. So, whatever epsilon you take whatever epsilon you take this delta will work because this inequality distance between x and p less than delta that is satisfied only by p only by p and no other no other point because there is no other point near x. So, there is no other point near p. So, if p is an isolated point then f is continuous at p. So, in particular for example, if you are set a it is such that every point is isolated then the function will be continuous at that point. For example, suppose a is n the set of all natural numbers then every point is isolated point there. So, any function defined on n will be continuous function. Next is suppose so let us so that disposes the case of the isolated point. Let us now look at what if it is not an isolated point it must be a limit point. In case of limit point what should happen is that limit f is continuous at p then the limit of f x as x tends to p should be same as f p that is what this second this definition says. So, if p is a limit point if p is a limit point of a then f is continuous at p f is continuous at p if and only if or this is equivalent limit of f x as x tends to p is equal to f of f. So, let us again and of course, this does not need any other proof because this sentence limit of f x as x tends to p is equal to f p is basically same as what we have written here this is basically same as what we have written here. So, let us again take the suppose p see in order to talk about continuity we have to have that p must belong to a since p once p belongs to a p can be either an isolated point of a or a limit point of a there is no third possibility. If p is an isolated point of a f is always continuous there if p is a limit point of a then limit of f x as x tends to p must be same as f p that is the requirement for continuity. So, now suppose let us ask this question what is the what is the way in which f can fail to be continuous of course, let us let us forget about the case when p does not belong to if f is not defined at p obviously, we cannot talk about the continuity at all. But suppose that is not the case suppose p belongs to a if p is isolated point obviously there is no question of f failing to be continuous. So, if at all f can fail to be continuous it will happen only at a limit point and in what way it will happen there are two ways either that this limit does not exist at all either that this limit does not exist at all or that this limit exist, but its value is different from this. So, so let us again summarize what are the ways of in which a function can fail to be continuous first of all it can fail to be continuous only at limit points. It can never fail to be continuous at isolated points there are two ways in which this can happen one is that limit of f x as x tends to p does not exist that is one way secondly limit exist, but it is different from f of p it is different from f of p. Now, among these two types you can see that the second type is easy to handle suppose a limit exist and the value is different from f of p then we can say that we can redefine that function and change the value of f at p and make the function continuous there that is possible. So, that is why that kind of discontinuity that if a function is not continuous at the point we say it is discontinuous at that point and those points are called points of discontinuity of function and this the second type which I mentioned just now that is called removable discontinuity and the reason is obvious because that discontinuity you can remove by simply modifying the definition of a function at that point, but if the limit does not exist at all then you can do nothing whatever way you modify the definition of f at the point p still the function will remain discontinuous there. You can see one more thing here that though in giving this definition here we have said that function is defined at a and then we p belongs to a etcetera. You can see that in all this definition and whatever discussion we have done so far compliment of a has no importance at all whatever happens to the points outside a we are not bothered at all. So, here afterwards we can simply forget about those points and talk of function going from x to y that is I just regard this a itself as a metric space I will regard this a itself as a metric space and talk about the functions going from a to y or which is same as saying x to y and we shall now give a very useful criteria for the continuous functions and let us say that now this time I am going to talk about function which is continuous everywhere. Of course, we can also give a similar description for the function which is continuous at a point, but it unnecessarily complicate the things. So, let us talk of this and that is in terms of open sets as we know open set is a very important concept in metric spaces and if we can talk of something purely in terms of open sets then that concept can be translated to topological spaces also because you know that we have defined what is meant by topological space in a topological space there is no concept of distance, but you have a concept of open sets. Now, here we have given a concept of continuity first using distance, but suppose it is possible to give the definition using only open sets then we can talk of functions continuous in topological spaces also that is the idea. So, let us let us take that first. So, I will write that as a theorem and also it is useful because using this equivalent criteria certain other proofs also become simpler. So, let us again say that x, y, rho are metric spaces and f from x to y is a function f from x to y is continuous this time I shall write continuous on x, but this is one convention suppose nothing is said if a function is continuous at a point we should say continuous at that point, but suppose nothing is said simply said f is continuous then it is assumed that it is continuous on x that means it is continuous at every point of x. So, f is continuous on x if and only if this is interesting what it says is that if you take an open set in y and look at its inverse image in x then that should also be open in x that is f is continuous if and only if inverse image of every open set is open. So, if and only now f inverse g is open in x for every open set g in y in other words this means that if g is an open set in y f inverse g is open set in x. Let us see how we can prove this now we can observe one more thing even before going to the proof of this coming back to this discussion of this continuity here you can say that saying that this d x p less than delta this is same as saying that x belongs to the open ball with centre at p and radius delta this last thing means. So, this means x belongs to open ball with centre at p and radius delta. What does this mean this means f p f x belongs to this means f x belongs to the open ball with centre at f p and radius epsilon. So, this means f x belongs to open ball with centre at f p and radius epsilon that is what it means that whenever x is in this ball f x is in that ball is it same as saying that the image of this ball that is f of this whole ball is inside this ball. So, this last sentence. So, this whole sentence whatever we write here that can be simply like this f of u p delta is contained in u f p epsilon. So, saying that f is continuous as limit point means for every epsilon there will exist a delta such that f of that open ball with centre at p and radius delta should be contained in open ball with centre at f p and radius epsilon we shall make use of this in the in this proof. In order to prove this let us first use this way suppose f is continuous suppose f is continuous then we want to say that if g is open in y f inverse g must be open. So, suppose f is continuous suppose f is continuous on x and g is open in y we want to prove that f inverse g is open in x that is this is what we want to prove to prove f inverse g is open in x. Of course, if f inverse g is open there is nothing to be proved if f inverse this is this is trivial. So, this is true if so true if f inverse g is empty what is the meaning of f inverse g is empty that means no f x goes to g that is no point in x its image is in g. So, that is the meaning of f inverse g is empty in that case nothing to be proved. So, next assume that suppose f inverse g is non empty suppose f inverse g is non empty means what some point belongs to it suppose f inverse g is non empty and let suppose that that p belongs to f inverse of g. Then we must show that p is a interior point that means that we should show that there exist a ball with center at p and some positive radius such that that ball is completely inside f inverse g. But p belongs to f inverse g means what it means f of p is in g. So, this means this means f of p is in g but this is an open set. This is an open set. So, there must exist some ball with this as a center which is completely contained this suppose I call that radius of that ball as epsilon. So, this implies there exist epsilon bigger than 0 such that open ball with center at f p and radius epsilon is contained in g. Now, we assume that f is continuous we assume that f is continuous at every point. So, in particular the point p also. So, for this epsilon there should exist some delta such that whatever it should happen. So, since f is continuous since f is continuous at p there exist delta bigger than 0 such that such that let me such that distance between x and p less than delta implies distance between f x and f p less than epsilon. We have seen that means this last that means whenever x belongs to u p delta f x belongs to u p epsilon or which is same as saying that there exist delta bigger than 0 such that I will write this f of u p delta this is contained in u f p epsilon and this is contained in g u f p epsilon is contained in g. Now, so what we have proved that f of this open ball f of this open ball is in g. So, does this mean that this open ball is in f inverse g because what we have proved that if you take any point in this ball if you take any x in this ball f x is in g. So, that means this ball is contained in the inverse image of g. So, this implies u p delta is contained in f inverse g. Now, that means that p is an interior point that means that p is an interior point and we have showed that every point p is an interior point. So, that shows that f inverse g is open. So, therefore f inverse g is open is it clear. So, we have shown that if g is an open set then f inverse g is also an open set this should happen if f is continuous on x. Now, we want to show that the converse is also true. So, let us take it this way now assume that the function f has the property that whenever g is open in y f inverse g is open in x then we want to show that f is continuous on x. To show that f is continuous on x means what we must show that f is continuous at every point in x that is. So, let us take any point suppose I call that point p. So, let p belong to x and let us take epsilon bigger than 0 let p belong to x and epsilon bigger than 0 and to show that f is continuous at p for this epsilon we have to find some delta. Now, see this u f p epsilon is an open set in y open ball with center at f p and radius epsilon this is open in y. So, it is inverse image must be open in x that is what we assumed if you take any open set in y it is inverse image must be an open set in x. So, therefore, we select f inverse f inverse of that u f p epsilon is open in x. Now, does p belong to this set it does because f of that is f p f of that is f p and it is it is in this ball. So, it certainly belongs to so and p belongs to f inverse u f p epsilon. So, remember this simply means that f of p belongs to this ball right that is true whenever you said p belongs to f inverse of any set it simply means that f of p belongs to that set and what is that set it is nothing but the open ball with center at f p. So, obviously f of p belongs to that all right. Now, p belongs to this set and it is open. So, what does that mean it again it should mean that there exist some positive number such that open ball with that p as a center and that positive number is completely inside this. So, let us call that positive number as delta. So, therefore, there exist delta because then 0 such that open ball with center at p and radius delta is contained in this f inverse of f inverse of u f p epsilon. And this is same as saying that f of u p delta is contained in u f p epsilon and that this is this means that whenever distance between x and p is less than delta distance between f and p is less than delta distance between f x and f p is less than epsilon right that is that is same as saying that f is continuous at p and since p was any arbitrary point in x f is continuous everywhere in x. So, as I said this describes the continuity completely in terms of open sets right if you look at this has nothing to do with the distance. Of course, in matrix spaces open sets are defined in terms of distance that is ok, but suppose you have some idea of defining open sets without using the distance then in that kind of spaces you can talk of what is band by continuous function by simply taking this as the definition. And that is what is done in topological spaces and that is what you will learn in your course in topology. Now, let us see how these things make certain proofs also quite simple like for example, now we have taken a function f from x to y suppose that is a continuous function and let us say we let me just write that also as another theorem again a fairly well done theorem. Suppose this time I take say three matrix spaces let us say x d y rho x d y rho and let us say z let us say some matrix eta here or matrix spaces actually see remember let me also say one more thing here when it is understood which matrix you are talking about or whether the particular reference to matrix is not important when simply says x is a matrix space right x is a matrix space or y is a matrix space strictly speaking one should say x d is a matrix space, but if it is understood which matrix you are taking or if the actual reference to the matrix is not important for discussion then it is quite customary to say that x is a matrix space. So, similarly I should have simply said x y z are matrix spaces and suppose f from x to y suppose f from x to y and g from y to z are continuous then we can think of a function which goes from x to z which is a composition of these two functions. So, define h from x to z by h of x is equal to g of f of x g of f of x g of f of x or which is usually denote as this is simply usually described as h is equal to g composed with f then h is continuous that is what we want to say then h is continuous. In short what we want to say is that the composition of two continuous functions is again a continuous function and instead of writing the proof in its detail I will just give you an idea and then we shall stop with it. See the idea is simply this we shall use this criteria we shall use this criteria to show that the function is continuous we want to say that h is continuous. So, h goes from x to z h goes from x to z. So, it is sufficient to show that if I take some open set in z then show that it is inverse image in x is open in x. So, I think it is better to explain this to the driver see f goes from x to y and g goes from y to z we are taking let us say we take some open set we take some open set g in z. We want to say that h now this composition is nothing but h this g composed with f is nothing but h. Now what we want to say is that h inverse of g which will be a set in x that is open that is open but what is the argument this g inverse of g is open in y then f inverse of the g inverse of g is open in x because f and g both are continuous but then all that you need to observe is that that is it is nothing but same as h inverse of g. In other words what you have to observe is simply this h inverse of g is f inverse of g inverse of g and if g is open here since g is continuous small g is continuous g inverse g is open in y and since f is continuous f inverse of this is open in x and that is same as h inverse of g. Now this is nothing but elementary set theory because h since if h is defined like this you can easily show that h inverse is nothing but f inverse composed with if h is g composed with f h inverse is f inverse composed with g inverse that is fairly elementary set theory. And so using that and this theorem we can show that the composition of two continuous functions is also continuous it can also be proved by using the usual epsilon delta definition. I suggest to you that you take that as an exercise try to prove it also without using this criteria and you will understand a difference we will stop with that.