 Now, see how mathematics builds up. So, let us write. Now, what is the meaning of node? Saying x minus c less than delta implies f of x minus f of c less than epsilon is equivalent to saying this is in terms of absolute value. I want to write it in terms of sets. It says, if I look at the interval, so look at the interval c minus delta to c plus delta. If x satisfies this property, x is in this interval, so x belonging to this implies f of x is where f of c minus epsilon f of c plus epsilon. That is same as saying, if I look at the image of the interval c minus delta to c plus delta, that is a subset of f of c minus epsilon to f of c plus epsilon of this interval. I am just rewriting everything slowly, changing the notations. f of x belongs to that. If x is in this, then f of x belongs, that means the image of this is inside that. So, what we are saying is, given a neighborhood of f of c, given an epsilon neighborhood of f of c, there is a delta neighborhood of the point c, so that the delta neighborhood is mapped into the epsilon neighborhood. Now, it is going in a set theory language. So, the reason is, in this language, it becomes, this definition of continuity becomes, is still in terms of intervals. If I write in terms of neighborhoods, given a neighborhood of the point f of c, there is a neighborhood of the point c, which is mapped into it. Then, everything is in terms of neighborhoods. Then, it becomes extendable. This definition becomes extendable, where there are no sequences, nothing but only neighborhoods. You play with neighborhoods. So, that is the interesting part of it. So, I just wanted to keep you, make you aware that these two are equivalent basis, only matter of saying what language you are choosing. For example, you can write this in terms of open sets. You can write in terms of open sets, because what is this? This is the neighborhood. So, supposing you are given an open set, which includes the point f of c, then there will be neighborhood inside it, and there will be neighborhood coming from inside this. What it says, that if you look at the inverse image of that neighborhood, that is also a neighborhood. That means, inverse images of open sets are open. That is another equivalent way of saying continuity. So, I am not doing much of this, because in your courses, probably you will not read it, but those who are interested and want to read later on, one can say, so it is equivalent. I am just saying for the sake of completion, otherwise not part of this course, as far as exam is concerned. So, do not bother about it. See, f continuous at x is equal to c, if for every neighborhood of f of c, every neighborhood u of f of c, there is a neighborhood v of c, say that f of v is inside u. Then one can write this in terms of open sets and so on. So, let us not bother much about it. So, what we have done is, we have looked at limits of functions, and then we looked at continuity property. Limit is, we treated limit as the value that you expect the function to take at points, considering estimating the value, by looking at the values of the function at nearby points. And when it is equal to actually the value of the function, we called it continuity, and then we looked at various properties of continuous functions. Here is something, which I think this is also good. So, we have two ways of defining continuity at x is equal to c. One was, for every x n converging to c, f of x n converges to f of c. And second, for every epsilon bigger than 0, there is a neighborhood delta bigger than 0, such that x minus c less than, x minus c less than delta implies f of x minus f of c is less than. Now, again I am stressing the point that continuity is the property of the function at a point, is a local property. What I mean by local property? Because, I was saying x n converging to c. So, specializing what is happening or even here, if for the same epsilon, supposing there are two different points of continuity c 1 and c 2, what is this for the point c 1, given epsilon some delta may work, that may may not work for other places. So, neighborhoods may change as the points change. Given neighborhood, existence of neighborhoods may change, at one point a bigger neighborhood may be ok, at other point may need to go for a smaller neighborhood. But there is a notion of continuity, which says irrespective of where you are, it works. So, let us define what is called definition f. Now, this is going to be a property, which is not a local, but more of a global property. f is defined on a domain D to R, we say f is domain D, I should write a interval, is uniformly continuous on D, if it is something like continuity, but it does not depend on the point. If for every epsilon neighborhood there is a delta, such that whenever x 1 and x 2 are any two points less than a distance delta should imply f of x 1 minus f of x 2 is less than epsilon. So, now it says take any two points x 1 and x 2 in the domain, if x 1 is close to x 2 by distance delta, it does not matter where they are, then f of x 1 is close to f of x 2 by the distance epsilon. If supposing I fix x 2, supposing I fix x 2 and is a point of continuity, then given epsilon there is a delta, such that this will happen, if x 2 is a point of continuity. Then for every x 1, the other distance at the most delta from x 2, f of x 1 minus f of x 2 will be less than. But this choice of delta, there existed delta may depend upon what is the point x 2. Supposing I reverse the rules, I say x 1 is a point of continuity, then again given epsilon there will be delta, say that x 1 minus x 2 less than will imply that. But that delta may be different from the earlier one when you are using continuity at the point x 2. So, let us, I will give examples to illustrate that. So, in a sense it looks like that given epsilon there is a delta does not seem to depend on the point where you are looking at continuity. Anywhere if two points are closed then the values are closed. Continuity says if the points are close to that given point then the values are close to the values of that point. So, that specializes on. So, let us look at probably some examples. So, that the simplest example is if you look at f of x equal to x is uniformly continuous. There is nothing because the function is not doing anything. It is not changing at all. If two points are closed f of x is x itself. So, nothing. But let us go a step further. Let us look at f of x is equal to x square. We know it is continuous at every point. f of x is continuous because f of x is continuous product of continuous function is continuous. So, f of x square is continuous if you want to look at it. But every polynomial function is continuous by that limit theorem is continuous. Let us try to write x is equal to 0 continuity. Continuity at the point x is equal to 0. So, what does it mean epsilon given? I have to choose there is a delta bigger than 0 such that x minus 0 less than delta should imply f of x minus f of x should be less than epsilon. That is continuity. So, let us just elaborate this for our setting. That means mod x less than delta should imply x square at 0 value is 0 is less than epsilon because f of x is x square. So, what is the best possible choice of there is a delta? So, what is the best possible selection of delta largest possible I can make? So, we can choose delta. What would you like to choose? Delta such that delta square choose delta such that delta square is less than or equal to epsilon. We have epsilon is given to you. So, choose obviously from this equation. Now, let us look at x is equal to 2. What is happening? At x is equal to 2. So, I want the same thing given epsilon bigger than 0. There is a delta bigger than 0 such that x minus 2 less than delta should imply f of x minus f of 2 less than epsilon. So, what is f of x? That is mod f of x minus f of 2. What is that equal to x square minus f of 2? The value is 4. So, this less than epsilon. So, epsilon is given to me. I have to choose a delta. So, what delta you choose? How do you choose delta? If you have not done this kind of a thing earlier, here is the way. See, I want x square minus 4 to be made small. What I know is x minus 2 is small. So, somehow I had to bring in x minus 2 in this. So, what we do is x square minus 4. I can write it as x minus 2 into x plus 2 mod of that. Then, whatever my delta is going to be, so this is going to be less than. It is only what we want to do less than delta into mod of x plus 2. If you still want, now still x is hanging around. I do not want that x 2. So, what you can do is you can write this. There are many ways one can proceed x minus 2 plus 4 because I want in terms of x minus 2, which is less than or equal to delta mod of x minus 2 plus 4. So, that is less than or equal to delta, no bracket here, delta this is less going to be less than delta plus 4. So, if I want x square minus 4 to be less than epsilon, I know this is going to be less than this. If x minus 2 is going to be less than delta, so choose delta such that delta into delta plus 4 is less than epsilon. Then, this quantity will become less than epsilon. So, everything will be okay. Now, you see what is the difference coming? Earlier given epsilon, I could choose delta square less than epsilon. Now, I have to make delta square plus 4 delta to be less than, same delta is not working. I have to make some more modifications. Geometrically, if you want to look at, here is the geometric way of looking at it. My function is y equal to x square. So, let us look at the graph of this function. This is the graph. So, at x is equal to 0, given epsilon, so let us say this is epsilon is equal to 1. Then what is my delta square? Delta square should be less than or equal to 1. So, that will work. So, between this and this minus 1 to plus 1, everything will be going there. But, if I look at some point here, so I have to do it 2 and I look at same epsilon equal to 1. That means what? This is 4, this is 3 and this is 5. That big thing will not work. I have to make my interval smaller, so that the values go inside it. So, what will be the value? Square root 3, other way around, 2 square root of 5. So, that interval I should take, then the values will go inside when you square it. So, it becomes different delta. For the same epsilon, different delta is required. So, I am just trying to illustrate with some example. But, what it is saying, uniform continuity says, I should not bother about where the point is. If two points are closed, then the distance between them is closed. That is called uniform continuity. Let us formulate this in terms of, one can ask, can I characterize this in terms of sequences? This is more like an epsilon delta kind of a definition. Sequences are much easier sometimes to handle. So, here is a theorem. So, f i to r, the following are equivalent. f is uniformly continuous and second. See, in continuity, we are taking a n converges to c. Then, f of a n converges to f of c. But, if we take two different sequences converging to c, then the limit also is converging. But, if I want to remove that point c, then what is the property? a n converging to c, b n converging to c. Then, what is happening to a n and b n? They are going to 0. What should happen? f of a n minus f of b n should also go to 0. So, that makes it independent of the point. So, what we are saying is, continuous is equivalent to saying for every a n b n belonging to i, mod of a n minus b n going to 0 should imply f of a n minus f of b n goes to 0. For example, if I take b n all equal to the constant sequence c, then a n converging to c implies f of, is it okay? So, obviously, says uniform continuity implies continuity. It is stronger than continuity. Is it okay? That uniform continuity is stronger than continuity. For example, if this theorem is true, you can look at this way or epsilon delta either way. I can take all the b n's to be the constant sequence c, then a n converging to c implies f of a n minus f of c goes to 0. That is continuity at the point c.