 So, we were discussing the properties of continuous functions. Let us recall that we had proved the following that if you take a function f from x to y, then f is continuous if and only if inverse image of any open set, inverse image of every open set in y is an open set in x and that is equivalent. And this is something quite useful in deciding about the continuity of a function and also proving several properties of the continuous functions. Is it clear to you that from this it follows that if x is a discrete metric space, then every function on it is continuous. Is it clear? Because whatever open set you take here, its inverse image, whatever be the set in x, since in x every set is open, so any function defined on a discrete metric space is always continuous. Now, let us also record a few more things that will follow from this. Now, let us say suppose you have two functions, suppose g from let us say g and f, let us say r continuous, then now this for this let us take this metric space as r. Suppose f and g are real valued functions and suppose both of them are continuous, then we want to say this that f plus g is continuous, f g is continuous. And if g of x is not 0 everywhere in x, then f by g is also continuous and here by continuous means continuous on x everywhere. Of course, we can also write a theorem at a particular point. So, if g x is not equal to 0 for every x in x, then f by g is continuous on x. As I said, here I have standard theorem for continuous on x, but you can as well state it for continuity at a point. Instead of say on x, if I change this that suppose both are continuous at some point, let us say point x not in x. Suppose f and g are continuous at a point x not in x, then f plus g is continuous at that point x not, f g is continuous etcetera. And the last thing will be if g x not is not 0, then f by g is continuous at that point x not. Now, as we have seen as far as the proof of any of these things are concerned, it will follow from two things. You just make two cases that to show that function is continuous at all points in x, just take any arbitrary point. Suppose that point is let us say x not, if that is an isolated point, then it is already continuous, there is nothing to be proved. If it is a limit point, then the required conclusion will follow from the corresponding theorem about the limits. For example, we have already proved that limit of f x f plus g of x as x goes to x not that is same as for example, that is for example, we have proved that limit of f x that is f is continuous at x not, f is continuous say if x not is a limit point saying that f is continuous at x not means this. Similarly, g is continuous at x not will be limit of g x as x to x not is g of x not and then by the corresponding theorem of the limits, it will follow that limit of f plus g at x not is same as f x not plus g x not and that will show that f plus g is continuous at that point x not. Since x not was arbitrary on the whole of x and similarly other two things also can be proved. So, the point is conceptually there is nothing new in this, all these assertions will follow from the corresponding assertions about the limits. Then we can also say one more thing here suppose we take let us say k functions let us say f 1, f 2, f k suppose these are k functions from x to r. Suppose each let us I will talk of continuity for the time being let us just say these are real value functions then I can construct what is called a vector value function that is function which goes from x to r k. Suppose I define that function as f, f is a function define f from x to r k by f of x is equal to f 1 x f 2 x etcetera f k x then f will be a function from x to r k given any if you take k real value functions it will lead to a function f from x to r k. Similarly, other way also if you take any function x any function f from x to r k that will give rise to k real value functions. So, just a similar thing as we have seen in the case of sequences now what we want to say is the following if all of these functions are continuous then f is continuous and conversely if f from x to r k is continuous then each of these must be continuous by the way these are sometimes called coordinate functions given by this function f. So, assertion of this f is continuous 1 x of course, here when I say r k I should say what is the metric on r k, but actually it does not matter whichever metric you take this theorem is true, but to just make the just to make the things concrete let us take the metric given by let us say in norms of x 1, but the same more or less the same proof will work for any other metric or any other metric given by any of those L p norms f is continuous on x if and only f j is continuous on x for each j and to see this all that we need to see is the relationship between the absolute value in r and the norm in r k. Since, I have already used this small x for the points in x here let me use something else for the point in r k. So, suppose I take a point let us say a in r k. So, a is a 1, a 2, a k. So, what we know is the following that is norm of a suffix 1 norm of a suffix 1 this is nothing, but sigma mod a j j going from 1 to k and so if so in particular each of this mod a j. So, each of this mod a j is less than or equal to norm of a each of this mod a j is less than or equal to norm of a. So, in particular see so how this how will this proof go suppose I want to suppose we assume that f is continuous on x. So, f is continuous at some x naught let us say suppose f is continuous at some then let us say we want to put that each of these are continuous at x naught. So, just look at f j at x naught minus f j of x essentially what we will use is this we shall say that if suppose you look at f j of x minus f j of x naught or let us say let us say f f of x minus f of x naught this will be an element in r k f of x minus f of x naught this will be an element in r k take that as this point a suppose I apply this then what will happen that is j th coordinate of this j th coordinate of that will be f j of x naught minus f at x naught that should be less than or equal to norm of f x minus f x naught now and let us also say what is and what is this norm of f x minus f x naught this is nothing but sigma j going from 1 to k mod f j x minus f j of x naught. So, once you realize this inequalities the proof of this is immediate for example, suppose f is continuous on x let us say f is continuous at x naught for each x and we want to show that each f j is continuous at x naught. Now, what is the meaning of saying that f is continuous at x naught it means given any epsilon given any epsilon you can find a delta such that whenever distance between x and x naught is less than delta distance between f x and f x naught is less than epsilon. That means whenever distance between x and x naught is less than delta this is less than epsilon, but if this is less than epsilon this also should be less than epsilon because f because this quantity is less than or equal to that. So, that shows that f j is continuous and that will hold for each j and what will be the reverse argument? Suppose, we assume that each f j is continuous then again similar thing which we have done that is if each of these let us say if each of these can be made less than let us say epsilon by k then the sum will be less than epsilon. So, we can say if each f j since each f j is continuous we can always find a delta such that whenever distance between x and x naught is less than delta mod f j x minus f j of x naught is less than epsilon by k. This we can do for each k this we can do for sorry this we can do for each j and then if each of this is less than epsilon by k the sum is less than epsilon. So, that will mean that norm of f x minus f x naught is less than epsilon. So, basically the whole thing follows from this and once you realize this you will see that if I take some other norm here instead of let us say norm of f x 1 suppose I have taken norm of f x 2 or non of f x infinity whatever you do this similar argument will work. Now, let us also recall one more thing we have seen that a function is continuous this is this implies that if you are take let us let us let me start from this suppose let me take any other way suppose f from x 2 y is a continuous function. Then we know that it from this it follows that if you take any convergent sequence in x its image in y must be convergent sequence because we have seen that equivalent of two definitions of limit. So, f is continuous means for any let us for any x naught limit of f x as x tends to x naught is f x naught. So, if you take any sequence x n converging to x naught f of x n must converge to f of x naught. So, we can say that f from x 2 y is continuous this is actually if and only if for if x n for every convergent sequence in x let us say sequence x n in x converging to converging to let us say x naught in x f x n converges to f x. So, continuous maps for the property that image of every convergent sequence in x is a convergent sequence in y. Now, let us ask next question what can we say about Cauchy sequences can we say that image of every Cauchy sequence is also a Cauchy sequence. To answer this question best thing is we will look at an example I will take x as the open interval 0 to 1 with the usual matrix and say I will take y as r then define f from x 2 y by f x is equal to 1 by x is that a continuous function that is a continuous function. Suppose I take x n is equal to 1 by n that is a sequence in x of course, it is not a convergent sequence it is not because 0 is not in x it is not a it does not converge to any point in x, but that is not important is it a Cauchy sequence it is it is a Cauchy sequence what is f x n f x n is n f x n is n is that a Cauchy sequence in r it is not right it is not. So, what does it mean in general image of a Cauchy sequence under a continuous function need not be a Cauchy sequence though we know that image of a convergent sequence under a continuous function will always be a continuous function image of a Cauchy sequence need not be a Cauchy sequence. Basically, to take care of this kind of things and of course, for few other things we now discuss a little stronger type of continuity that is called uniform continuity. Now, even before coming to this definition uniform continuity let me also introduce one or two more terminologies more definitions let me remove this for the time. Suppose, f x and y are two matrix spaces and f from x to y is continuous let us say suppose in addition f is also a bisection suppose f is a continuous and bisection. Bisection means it is it is 1 1 and on to and we have known that if a if a function is a bisection we can define what is meant by inverse function. So, then we can define then f inverse is a function from y to x, but f inverse may or may not be continuous if f is continuous and bisection inverse function exists, but at inverse function may or may not be continuous in general unless we put some additional conditions on x y or f etcetera. But, when it is also continuous that map f is called a homeomorphism. So, we will say that f is called a homeomorphism homeomorphism if f from x to y is continuous it is bisection and f inverse from y to x is also continuous. We can give obvious examples of homeomorphisms for example, an entry map going from x to x that is continuous it is inverse is also continuous also we can also construct a non trivial example that is not very difficult. Let me give you as an exercise construct an example of a function which is continuous and bisection, but whose inverse is not continuous. So, let me just give it as an exercise. So, give an example of f from x to y that is continuous bisection, but and f inverse is not continuous that means it is continuous bisection, but not homeomorphism f inverse from y to x is not continuous. Is this clear what is meant by homeomorphism what should happen is that the function f should be continuous inverse should exist. So, it is a bisection and inverse also should be continuous. Now, if you are given two metric spaces we say that those two metric spaces are homeomorphic to each other we say y is homeomorphic to x if there exists a homeomorphism from x to y. So, given we say that y is homeomorphic to x y is homeomorphic to x if there exists a homeomorphism from x to y of course, there is nothing unique about homeomorphism given to metric spaces there may I mean there may not exist any homeomorphism there may exist several homeomorphisms also. Now, this establishes a relation between the set of all metric spaces if you take the class of all metric spaces this says how two metric spaces are related to each other is this is it clear to you that this is an equivalence relation x is homeomorphic to itself that is clear identity map is a homeomorphism if x is homeomorphic to y does it follow that y is homeomorphic to x that is clear if f is a homeomorphism going from x to y f inverse is also a homeomorphism going from y to x what about transitivity suppose we know that x is homeomorphic to y and let us say y is homeomorphic to z will it follow that x is homeomorphic to z obviously, because of what you do is that just take the composition of two homeomorphisms and show that the composition is also homeomorphism and to do that we shall use the theorem that we produce today that composition of two continuous maps is continuous all right. So, this is an equivalence relation and as we have seen earlier any equivalence relation will partition the given class into what are called equivalence classes. So, every metric space in that equivalence class will be homeomorphic to all other metric spaces in that equivalence class all right. Now, let us come to this uniform continuity now here we talk of uniform unlike we can talk of continuity at a point, but we talk of uniform continuity on a set uniform continuity on a of course in particular the set can be a singleton set, but that will be trivial. So, let us just take this case suppose x and y are two metric spaces and as I said yesterday when the metric underlying matrix are not very important for discussion I will not say explicitly that x d is a metric space or y rho is a metric space etcetera whenever that is required we will we will see that for the time being let us let me take this let us say x d and y rho are metric spaces and suppose f is a map which goes from x to y f is a map which goes from x to y and suppose let us say it is continuous suppose x to y is continuous. Now, let us recall the definition of continuity here once again what we have said in the definition is that is that given any epsilon bigger than 0 that is f is continuous means f from x to y is continuous means f is continuous at each x f is this means f is continuous f is continuous at x for every x in x for every x in x f is continuous at x. Now, what is the meaning of this it means that that is for every epsilon bigger than 0 for every epsilon bigger than 0 there exists delta bigger than 0 such that for all y in x distance between y and x less than delta implies distance between f y and f x is less than epsilon. Now, the whole idea of uniform continuity deals with how does this delta depend on epsilon we know that delta may depend on epsilon if epsilon is changed delta is to be changed, but it is not made explicitly clear the definition of continuity that the delta will depend on this x also in general. For example, suppose I take some other point is x 1 or x 2 then it is not it is not necessarily the same delta will work for that x also. So, if it so happens that you can find some delta which works for all elements in some set which works for all elements in some set or in particular for the whole of x then we say that f is uniformly continuous on that set a and if a is equal to x we say f is uniformly continuous on x. So, let us say that let me just get this definition. So, suppose now a is a subset of x f is said to be uniformly continuous on a uniformly continuous on a if for every epsilon bigger than 0 there exists delta bigger than 0 such that this delta will work for all elements in a this delta will work for all elements in such. So, whenever you take any x in a and any y in x right. So, what we say this is important such that for every x in a for every x in a and let us say y in x distance between x and y less than delta this implies distance between f x and f y less than x. Remember once again uniform continuity is always discuss with respect to a set we always say a function is uniformly continuous on a set. Whereas, continuity we can say that function is continuous at a point or on a set both are possible and of course, the special case is when a is equal to x when a is equal to x then f we say f is uniformly continuous on x all right. Of course, a can be a singleton set, but it only means that f is continuous at that point and if a has only one point obviously whatever delta works for that point it works for all points in that set a. So, if a is only one point then any function continuous at that point is also uniformly continuous on a is it also clear that with a small modification of this we can also make this work if a is a finite set. Suppose a is a finite set if there are finite number of points given any epsilon we can find let us say there are let us say there are n points a 1 a 2 a n. So, for each of those a 1 a 2 n you can find delta 1 delta 2 delta n suppose delta 1 works for a 1 delta 2 works for a 2 etcetera delta n works for a n then what you do obviously you take the minimum of all those deltas you take minimum of all those deltas that will work for that will work for every point in that set a. So, what we have essentially set just now is that every function which is continuous it will always be uniformly continuous on a finite set every continuous function will always be uniformly continuous on a on a finite set. So, that is not very important. So, really uniform continuity really matters when we take a as an infinite set and that may or may not happen a function may be continuous, but it may not be uniformly continuous. Let us just see some examples. So, that these ideas will be clear. So, let me start with familiar spaces. So, let us say f from R to R by the way let me again go back to our favorite space discrete metric space we already seen that every function define on a discrete metric space is continuous is it also clear that it is uniformly continuous also. Suppose suppose x suppose x is a discrete metric space then what I can do is that whatever be the epsilon given I can take this delta to be half suppose you take this delta to be half then d x y less than delta can happen only when x is equal to y and in that case distance between f x and f y will be just 0. So, on a discrete metric space not only that every function is continuous, but it is also uniformly continuous. Now, let us take this space and let us look at this function f x is equal to 2 x we know that this is a continuous function the question is whether it is uniformly continuous or not. So, to do that let us look at say take some epsilon let epsilon be bigger than 0 then we have to find delta such that whenever mod x minus y is less than delta mod f x minus f y should be less than epsilon. So, mod f x minus f y this is nothing, but mod 2 x minus 2 right. So, this is nothing, but 2 times mod x minus y and we want to choose delta in such a way that whenever mod x minus y is less than delta this should be less than epsilon. Now, the choice here is obvious you take just delta is equal to epsilon by 2 delta is equal to epsilon by 2 and this delta will work regardless of what are x and y as soon as mod x minus y is less than epsilon by 2 mod f x minus f y is going to be less than epsilon. So, this is an example of a uniformly continuous function right. So, f is uniformly continuous on r is uniformly continuous on r. Now, let us see some other function which is not uniformly continuous. Now, before proceeding further let me give you an exercise on your own show that sin x is uniformly continuous on r. Let us take another function g from r to r suppose I take g of x is equal to x square. Now, let us try to do the same thing that is given epsilon bigger than 0 try to find delta satisfying the requirement. So, let epsilon be bigger than 0 then what is g of x minus g of y. So, if you look at g of x minus g of y absolute value of this that is nothing but mod x square minus y square that is nothing but mod x square minus y square and this is nothing but we can write this as mod x minus y into mod x plus y. Now, what we want when mod x minus y is less than delta mod x minus y is less than delta this whole thing should be less than epsilon. Now, how do we choose delta like that? Yeah, but y is something that will depend let us say you see what we want is this mod x minus y less than delta this should imply mod x minus y is mod x minus y into mod x plus y this is less than epsilon. So, what we have to do is that let us see that whenever mod x minus y is less than delta what is the maximum possible value of this factor mod x plus y. So, we can say that mod x plus y we can say that this is I can write this as mod x minus y plus 2 y or better still let us say we are talking about continuity at x we can write it as mod y minus x plus 2 x. So, this will be less than or equal to 2 times mod x plus delta 2 times mod x plus delta. So, if I take if I if I choose. So, this will be what I want is delta into 2 times mod x plus delta this should be less than epsilon this should be less than epsilon. Now, we can see the idea the one the way in which one does this is that see we know that if any particular delta works any delta smaller than that will also work any delta smaller than that will also work. So, I can also make another decision that I will choose delta smaller than mod x. Of course, to choose delta smaller than mod x we must require that mod x is not 0 fine we can make that if mod x is equal to 0 that means x is 0 for that we shall think of something else for that we shall think of something else that case we can discuss separately. Suppose, we take the case mod x is bigger than 0 then if I choose delta in such a way that delta is less than mod x then this whole thing will be less than 3 times mod x. So, that means I choose then we have to choose delta in such a way that delta into 3 times mod x is less than epsilon. So, that means delta should be less than epsilon by 3 times mod x that is the requirement and we also want delta is less than mod x. So, we can say that look at minimum of these two numbers minimum of mod x and 1 by 3 times mod x and take delta to see if mod x is bigger than 0 if x is not 0 this number is going to be strictly bigger than 0 choose any delta which is smaller than this that is 0 less than delta less than this that delta will work that delta will work. But it is clear from this discussion that this choice of delta depends on x choice of this delta depends on x if you change the point x the value of delta will change value of delta will change. So, we cannot the same delta will not work for any arbitrary x here. So, this function is not uniformly continuous this function is not uniformly continuous on R. But on the other hand suppose I take some close that boundary interval in R instead of taking the whole of R suppose I take a as simply 0 to 1 suppose I take a as simply 0 to 1 then what you can do is that take the maximum possible take this minimum over all x which belong to that and then take the minimum value of that and then we can find the delta which works for all elements here. So, it is not uniformly continuous on R, but it is uniformly continuous on sets like this it is uniformly continuous on sets like this. So, uniform continuity as I said earlier uniform continuity is a matter on depends on the set on which you are talking about uniform continuity if you change the set if function may be uniformly continuous on some set A it may not be uniformly continuous on some other set B. Now, let us go back to the whole point we started with we said that given an arbitrary continuous function image of a convergent sequence is always convergent, but image of a Cauchy sequence need not be Cauchy. Now, does uniform continuity corrects that view that is can we say that if a function is uniformly continuous then image of a Cauchy sequence is again a Cauchy sequence. Now, before going to that question let me again give you an exercise the same function which we discussed take x is equal to 0 to 1 and y is equal to R and f of x is equal to 1 by x check whether f is uniformly continuous this is an exercise check whether f is uniformly continuous. Now, coming back to that question so suppose f from x to y is uniformly continuous uniformly continuous and x n is equal to is a Cauchy sequence in x x n is a Cauchy sequence in x f x n is a Cauchy sequence in y. So, this shows that under uniformly continuous function image of a Cauchy sequence is a Cauchy sequence anyway how does one show that something is a Cauchy sequence just go by the definition take epsilon bigger than 0 then you have to find some n 0 such that whenever n m and n both are bigger than equal to m 0 distance between f of x n and f of x n should be less than epsilon. So, let epsilon be bigger than 0 then for this epsilon we will have to find n 0 satisfying those requirements, but we know that f is uniformly continuous and x n is a Cauchy sequence. So, these things we should use these things we should use first of all f is uniformly continuous that means for this epsilon there will exist some delta such that whenever distance between the two points here is less than delta distance between their images here is less than epsilon. So, because since f let me now give some matrix also x d and y rho since f is uniformly continuous continuous of course I mean uniformly continuous on x uniformly continuous there exist delta bigger than 0 such that f is uniformly continuous for any two points x and y for any two points x and y in x distance between x and y less than delta this implies distance between f x and f y is less than epsilon. And remember here that it is important to note that this delta does not depend on what is x and y this is important this part is important for every x and y in x this same delta works unlike that function there where delta will depend on the corresponding choice of x. Now we know that x n is a Cauchy sequence. So, what I can say is that for this positive number delta there will exist some n 0 such that whenever n and m is bigger than equal to that n 0 distance between x n and x m is less than delta right. So, since x n is a Cauchy sequence in x since x n is a Cauchy sequence in x there exists n 0 in n such that m and n bigger than or equal to n 0 implies distance between x n and x m is less than delta. Now if distance between x n and x m is less than delta what about distance between f of x n and f of x m that should be less than epsilon because you take x as x n and y as x m here. So, this implies distance between f of x n and f of x m is less than epsilon. So, we have proved that image of a Cauchy sequence under a uniformly continuous function is again a Cauchy sequence which is this is a property which is not true for an arbitrary continuous function. So, uniform continuity is a stronger type of continuity obviously every uniformly continuous function is continuous, but the common is false we have seen the we have seen the examples. Now, let us see some common examples or popular examples of uniformly continuous functions. In fact, some of these types are also given certain particular names one of those is what is called Lipschitz continuous function Lipschitz continuous Lipschitz is again a common name of mathematician perhaps you may have heard of this Lipschitz continuity while discussing differential equations. So, again let us say that take two metric spaces x d and y rho metric spaces and suppose we take a function f from x 2 y function f from x 2 y this is called Lipschitz continuous this is called Lipschitz continuous on x this is called Lipschitz continuous on x if there exist some alpha bigger than 0 if there exist some alpha bigger than 0 such that such that distance between if suppose you take any two points x and y here the distance between f x and f y is that is not equal to this number alpha times distance between x and y. So, what should happen is that for every x and y in x. For every x and y in x distance between so rho f x f y is less than or equal to alpha times distance between x and y. Now, is it obvious that every Lipschitz continuous function is uniformly continuous because you can just take delta as epsilon by alpha suppose you take delta as epsilon by alpha whenever this is less than delta distance between f x and f y will be less than epsilon. So, every Lipschitz continuous function is uniformly continuous of course, it is natural to ask here what about the converse, but we shall not going to that question right now. Now, another special case of this if this constant alpha is strictly less than 1 then this map is called a contraction Lipschitz continuous function is called a contraction if alpha is less than 1. So, if it is called a contraction alpha is less than contraction or contraction map it is called contraction or contraction map. See suppose I had not defined this Lipschitz continuity at all and suppose I want to define contraction directly then what I should have said is that there exist alpha such that 0 less than alpha less than 1 and rho f x f y less than or equal to alpha times d x y for every x y in x that will be the difference of contraction. Here since we already defined what is Lipschitz continuous function contraction is a special case of Lipschitz continuous function and you can also see why it is called contraction. It is called contraction because the distance between the images will be strictly less than the distance between x and y. So, the distance is shortened that is why it is called it is called a contraction and I will just make one more definition and then we will stop with that. This is also something which you come across very often and it is what is known as isometric. Isometry is a map which preserves the distance f is called an isometry if distance between f x and f y is same as distance between x and y. So, f is said to be an isometry if distance between f x and f y is same as distance between x and y for every x y in x. So, all this distance Lipschitz functions, contractions, isometries all of these are examples of uniformly continuous functions and we shall come across the examples like this. It is also it is clear from the definition of an isometry that isometry will be always 1 1 that is clear because suppose f x is equal to f y it will mean that rho f x f y is 0 that will mean the distance between x and y is also 0 that will mean x is equal to y. So, isometry is always 1 1 always 1 1 it may or may not be on to isometry may or may not be on to, but if it is also on to we say that x is isometric to y. We say that x is isometric to y if there is a on to isometry if there is an on to isometry will always be 1 1 by definition if there is an on to isometry means it automatically becomes a bijection then we say that x is isometric to y and this also establishes relation between the two metric spaces just like homomorphism when we say x is homomorphic to y. So, similarly we say x is isometric to y if there is an isometry from x on to y this is also an equivalence relation will stop with that.