 So, we shall continue our discussion about the real valued functions defined on some subsets of the real length namely the intervals. So, in the last class we were discussing the meaning of symbols like this that is let us write it in this fashion f x tends to l as x tends to a and we have already defined what is the meaning of this when l and a both are real numbers. Now, what we want to do is that or rather what we have done in the last class is that we extend the meaning of this idea of the limit by allowing this l as well as this a to be in the extended real length. So, l and a can be both in the extended real length and remember that extended real length is usual real length in addition with these two symbols minus infinity and plus infinity. So, what has the additional thing that has happened is that now l can be a real number or plus or minus infinity and similarly a can also be a real number or plus or minus infinity. Now, let us recall what was the idea how we have defined what we had seen is that if either a or l is a real number then we take neighborhood of a that is that we take as something like a minus delta to a plus delta that is an interval containing a or a minus epsilon to a plus epsilon either one of those and neighborhood of infinity that we take as something like this m to infinity that is nothing but the set of all x such that x is bigger than m and similarly neighborhood of minus infinity that we take as minus infinity to m and so what is the meaning of this f x tends to l as x tends to a we say that this means let me again write f x tends to l as x tends to a this means that for every neighborhood u of l there exists a neighborhood v of a such that whenever x belongs to v f x belongs to u or which is same as saying that we will say that f of v is contained in u and when a and l both are real numbers we take this number who u as l minus epsilon to l plus epsilon and v as a minus delta to a plus delta and then our definition becomes for every epsilon there exists delta such that etcetera. So, similarly using these neighborhoods we can translate these definitions also in a very usual manner if you want let us just take one special case. So, suppose let us say a is real number and l is infinity that means f x tends to infinity as x tends to a and a is a real number. So, what will be the meaning of that for every neighborhood u of infinity neighborhood of infinity is like this m to infinity. So, it will mean that for every m bigger than 0 there exists now neighborhood of a is. So, v is a minus delta to a plus delta and u is m to infinity u is m to infinity. So, there exists delta bigger than 0 such that what should happen f of v is contained in u that means f of a minus delta to a plus delta should be contained in u that is mod x minus a less than delta. This means x belongs to v mod x minus a less than delta that means x belongs to v this should imply f x bigger than m. f x bigger than m means f x lies in this f x lies in this m to infinity that is the meaning. Similarly, you can rewrite the definitions by considering various combinations where a can be plus or minus infinity or l can be plus or minus infinity or one of those real numbers and other is infinity all sorts of possible combinations are possible. We shall not go into separate discussions of all these cases this one definition covers all those possibilities by just this understanding that neighborhood of a real number is of the form a minus delta to a plus delta or a minus epsilon to a plus epsilon and the neighborhood of plus or minus infinity are of this form. Now, using this definition we can also show that whatever are the usual theorems about the sums and products of limits etcetera that we have discussed they are also true in this new situation also. So, we shall not go into very detail proof of that I will just take those theorems here because those are some things which we shall be using in future. So, suppose f x tends to l and g x tends to m as x tends to a and here l m a all can be all are in extended real length that is l m a all are in extended real length. So, any of those can be plus or minus infinity or a real number then again as x tends to a first thing is f x plus g x tends to l plus a sorry l plus m and then f x into g x tends to l into m and last thing is f x by g x tends to l by m. And if you remember in the usual way of stating the theorem we stated these two things as they are only by stating the last theorem we stated that force if m is not equal to 0 that is because if m is 0 l by m is not defined but that is the that is the care now we have to take in all these cases because it can happen that l plus m is also not defined because we have seen that suppose you have l is infinity and m is minus infinity then this is infinity minus infinity that is also not defined. So, if here also if it is infinity into infinity into 0 that is also not defined. So, what I will say is that all these things are true provided the right hand sides are well defined that is all these things are true provided right hand side I will separate RHS is defined. We shall not discuss the proof of this theorem because there are no new ideas involved basically the same same same idea will hold here. Now, let us again let us go to now discussing new concept here new in the sense we are discussing it for the first time, but you would have already come across this in your under graduate courses namely differentiation. So, let us just recall a few things again here as I said when we are discussing the real valued functions it is not our aim to just repeat whatever you already learn during your under graduate courses what we want to do is that we shall just review a few things and try to emphasize where exactly the ideas of real analysis come into picture like things like connectedness compactness see what could have happened is that you might have used those ideas earlier without realizing that that is a thing that is involved. So, we shall point out whenever such things happens. So, let us begin with the definition. So, suppose f is a function and that is defined from a b to R and suppose we take some x now for the convenience let me say a less than x less than b then we will define let us say a new function let me call it phi phi of t we define that as f t minus f x divided by t minus x. Obviously, this cannot be defined if t equal to x. So, we will say that this is defined for all t in a b and t not equal to x or you can say t in a b minus x. So, now we consider limit of this function that is limit of phi t as t tends to x of course, phi is not defined at x, but that is not required for considering the limit that is what we have seen for defining for talking about a limit the function need not be defined at that point. So, we can talk about a limit of phi t as t tends to x if this limit exists and now when I am thinking of this definition I mean the limit exists as a real number now I am not talking of these extended definition of limit. Now, if this limit exists as a real number then we say that f is differentiable at that point x and whatever is that limit a real number that will be called a derivative of f and we will denote that by let us say denote that by f prime of x of course, there are many other notation d f by d t at t equal to x etcetera let us not bother about that right now for time being we will say we shall. So, let me say that if limit exists if limit of let me give this function in full f t minus f x divided by t minus x as t tends to x exists we say that f is differentiable at x and the value of the limit the value of the limit is called the derivative of x value of the limit is called the derivative remember we already proved that whenever limit exists it has to be unique there cannot be two different values of limit. So, with the limit exists there is only one unique value that is the value real number which will be called the derivative that is why the derivative of f at x and we will denote it by f prime x. Now, one well known thing is which about derivatives or differentiability that you have already learnt I am sure is that whenever a function is differentiable at a point it is continuous at that point. So, let us just recall that if f is differentiable at x then it is continuous at x it follows immediately by looking at the definition of the derivative that is recall that since the whole thing is happening inside an interval all points are limit points. So, when we say that the function is continuous what we will mean is that limit of f t as t tends to x must be f x that is what we require and to see that we just look at the difference f t minus f x and write that as f t minus f x divided by t minus x and then multiplied by t minus x and consider the limit of both sides as t goes to x we already know that since f is differentiable at x this goes to f prime x this goes to f prime x and t minus x obviously goes to 0 as t goes to x. So, this goes to 0 so the product will be 0 and so that means that limit of that is same as say that limit of f t minus f x as t goes to x is 0 which is same as saying that limit of f t as t goes to x is f x and that is same as say that f is continuous at x. Of course, this is something that you already know, but the point is we have made a crucial use of the fact that f is differentiable at x here. So, this proof will obviously not work if f is not differentiable at x, but of course, that immediately does not say that the converse is false. To show that the converse is false again as I said there is only one way we have to think of a counter example that is counter example of a function which is continuous, but not differentiable at some point again I am sure you would have come across that function it is a well known function namely the absolute value of x or mod x which is which is continuous in fact at all points, but it is not differentiable at x equal to 0 and one can use several examples like that there is no depth of examples. Then we shall also just quickly review the theorem about the various operations on derivatives. Suppose we take say two functions f and g defined on the interval and suppose both are differentiable at a point then we want to know what happens for the sum and product etcetera. Now, before that let me also say one more thing here here we have talked about limit of phi t s t goes to x and when we say nothing about the limit it is the usual limit, but we have also considered for the real valued functions what is meant by left hand limit and right hand limit etcetera it is possible to consider that also it is possible to consider that also and that sort of thing will be called left derivative and right derivative etcetera. So, it is quite and again from whatever we have learned about the limits it is well known that if the limit exists then both left hand limit as well as right hand limit will exist, but the converse is false that is it is possible that left hand and right hand limit may exist, but the limit may not exist which is what we have called a discontinuity of the first step namely jump discontinuity. So, using those ideas we can define what is meant by left hand derivative of a function what is or say we can say function is differentiable from left or differentiable from right and those values we can define as a left hand derivative and right hand derivative etcetera, but we are not going to do that kind of thing here. I mean it is possible to do that, but since no really new concepts are involved we shall not go into that there is only one thing that we note you should notice here. I have taken here a strictly less than x less than b that is because we are considering the usual limit if you take x is equal to a since there is nothing on the left of a. So, the only limit that you can consider is the right hand limit. So, at the point x equal to a if you want to talk about derivative it will be only right hand derivative. Similarly, at the point x equal to b it will be only left hand derivative. So, sometimes we may need to talk about that kind of thing at the end points of an interval. So, at that time we will mention, but otherwise we shall not go into too much details about the left hand and right hand derivative coming back to what I said just now. Suppose, we take two functions let f and g from a b to r and suppose f g are differentiable at x in a b. Again I will take x in this interior point x in just to avoid this. So, discussion of left hand and right hand derivative we shall take the interior point. Then first thing is f plus g is differentiable at x and the value of derivative that is f plus g prime at x is f prime x plus g prime x. You can see that there will be nothing much in the proof of this. This will follow simply by the corresponding theorem about the limit because if you write the for example, if you write the function of phi t replacing f by f plus g it will be f plus g of t minus f plus g of x divided by t minus x. So, it will be f t minus f x plus g t minus g x divided by t minus x and if the limit of both since f and g both are differentiable the limit of each of those components will exist. So, the limit of the sum will exist. So, all these things follow whatever I say about derivatives all these things will follow by using the corresponding theorems about the limits. And of course, some simplification and some adjusting of sums etcetera. So, second part is that about the product f into g is also differentiable at x at x and derivative of f g at x that so called product formula. So, that is f prime x into g x plus f x into g prime x. Again here also there is nothing much involved see if you instead of this f you replace f t f f into g everywhere. So, it will be f t into g t minus f x into g x and if you remember what we do is that you add and subtract a term f t into g x and then just since the limit is with respect to t wherever the terms involving only x are there you just take them as a common factor and then you will get this formula there is again there is nothing much in this. So, and finally, this f by g is differentiable at x and the derivative of that f by g prime at x f by g prime at x. Remember that this is different from the corresponding theorems on limits the product of the derivative of the product is not same as the product of the derivative unlike the limits, but you of course this follows by using the corresponding theorems on limits, but before that you have to do some work because if you replace f by f t into g t etcetera you are not getting the product of this kind of fractions. So, that is where you need some work. So, similarly here for f by g so if you replace this by f t by g t and this by f x by g x you have to simplify and adjust the corresponding terms and then use the corresponding theorems about the limits. We shall not go into that kind of details because I am sure you have done you have seen these kind of proofs in your under graduate courses. If you have forgotten you can always look into one of those books. So, what is this formula f by g whole prime x that is this is g prime x square then here you have f prime x into g x minus g prime x into x this is g prime x square yes sorry g of x whole square I am sorry it should be g of x whole square that is right. Now, coming to the usual functions and their derivatives again as I said this is a topic which you would have done quite thoroughly in the under graduate course. We shall not spend any time on that how to find the derivatives of some well known functions by using various formulas and doing things like so called logarithmic derivatives etcetera etcetera those type of things we shall not go into we shall not discuss in this course. But we shall just recall that we will need the facts that derivatives of some well known functions are again some well known functions let us just recall a few things. For example, if f is a constant function then its derivative is 0 that is again follows clearly from the definition and further if let us say f x is equal to x f x equal to x then that function phi t which will write that will simply become t minus x divided by t minus x. So, that will be just one constant function one. So, in this case this is differentiable everywhere and so f prime x will be equal to 1 for all x. Then using this fact and repeat using this second thing here we can write we can write suppose I take g x is equal to x to the power n then you have to just use this repeatedly x square x cube etcetera etcetera products of various thing using this repeatedly we can we can get that g prime at x is n into x to the power n minus of course, here I am assuming that n is a positive integer power. So, n is n of course, n equal to 0 is also fine if you take negative powers then you will have to use you will have to use this of course, I should have mentioned here that this last formula is very provided g x is not 0, but that is understood because at no point of time we are allowing the in the discussion of derivatives where we say derivative exists it means that it exists as a real number we are not allowing the extended real line value for the derivatives at least right now. Let us see if we need to change that in future. So, if you take n as a negative integral value then you will have to use this last thing also right. Then you can use this well known functions like things like derivative of e power x is e power x or log x is 1 by x and things like that and various trigonometric functions and inverse trigonometric functions and all those things. Let me just say one more thing here that or one theorem which is quite useful and which will be which needs to be used quite often and what is called chain rule. This is about the derivative of composite functions. So, again let us say that suppose f from a b to r is differentiable x in again let me take some open interval a to b. Suppose we consider one more function g that is defined not on this interval a b, but on some interval which contains the range of this function f on some interval which contains the range of this. So, let us say that let this range of this interval I shall note by f of a b. Let f of a b be content in some interval i g is defined from i to r g is defined from i to r and since f of this whole interval lies inside this interval i f of x is a point in i f of x is a point in i and so g is defined at that point g is defined at that point. So, g from i to r and suppose I assume that g from i to r is differentiable at f x at f x. Remember f x is a point in i and g is defined on i then what this chain rule says is that the composite function g compose with f that is differentiable at x and it tells us the value of the derivatives. I will continue here let us say define h from a b to r by let me write like this h is equal to g compose with f h is equal to g compose with f or if you want to write in the full form that let us say h of t is equal to g of f of t for t in a b h is differentiable at x h is differentiable at x and h prime at x that is the derivative of h at x it is same as g prime at f x and multiplied by f prime at x that is see till now we had just talked about the algebraic operations on the functions now this is a different type of operation here we take the composition of two functions and then this says when is the composite function composite function here is h is g compose with f if g and f are differentiable that is f is differentiable at x and g is differentiable at f x then h is differentiable at x its value is given by this formula this formula is the one which is called chain rule again I am sure you have seen this formula earlier and used it also I shall just discuss the main idea in proving this without going into too many details of this how how does one go about proving this what is done here is as follows see f is differentiable at x let us again recall the definition what we had done is that we had defined this function phi phi t which was defined as f t minus f x divided by t minus x this was defined for t not equal to x so what we can say is that what we know if f is differentiable at x this phi t has a limit as t goes to x as t goes to x and that limit is nothing but f prime x that limit is nothing but f prime x so which is same as saying that if I if I look at the function phi t minus f prime x if I look at the function phi t minus f prime x suppose I call that function something suppose I call that function let us say u of t then u of t is it clear that u of t tends to 0 as t tends to x saying that phi t tends if f is differentiable at x what we should have what we should have is that this function u of t should go to go to go to 0 as t tends to x now what I can do what I will now do is that I shall use this whole idea as follows I will say that f t minus f x is nothing but phi t into t minus x and phi t is nothing but f prime x plus u t so I can say that f sorry f t minus f x this is same as I will say f prime x plus u t into t minus x f prime x plus u t into t minus x and what is the all that I have done is just I have written this whole thing and with the property that u t tends to 0 as t tends to x now whatever I have done for f whatever I have done for f I can do the same thing for g only only difference is now g is not differentiable at x but it is differentiable at f x suppose I call that point f x is y then it is same as in a g is differentiable at y and just to avoid confusion instead of using the variable t I shall use some other letter let us say s so what I can say is that is just this f t minus f x let me call say g s minus g y then g x minus g y since g is differentiable at y if g is differentiable at y let me write this y is equal to f x we are assuming that g is differentiable at y then we can write g x minus g y similarly as this will be g prime y plus some function of s which goes to 0 as s goes to y which goes to 0 as s just as here we have taken u t there will be some function some other function suppose I call that function v of small v of s v of s into instead of t minus x this will be s minus y instead of t minus s it will be s minus y where what what we know is that where v s tends to 0 as s tends to y now we just use these facts in simplifying and getting this I shall just give one more step and maybe remaining part of the proof you can complete now we want to show that h is differentiable so we should look at the corresponding fraction of about h that is what we should look at is h t minus h x divided by t minus x that is what we shall see that is what we should look at right that is our aim is to show that h t minus h x divided by t minus x this has a limit as t tends to x all right this dividing by t minus x I will do little later now h t minus h x by by that this is say minus g of f t minus g of f x right let us just use this notation what is the one notation one notation is y is equal to f x and let us say s is equal to f t then this is nothing but g s minus g y is nothing but g s minus g y and use this g s minus g y is g prime y plus v s into s minus y that is this is nothing but g prime y plus v s into s minus y where what is the property of this v s v s goes to 0 as s goes to y and y means f x okay now let us write let us unpack this s minus y s is f t and y is f x okay so now I will change these two is it clear that I will I will rewrite this okay so this is I will write this as s is f t and y is f x okay now use this first thing that is f t minus f x is equal to f prime x plus ut into t minus x okay so this is same as so this will remain as it is this is g prime y for this y I will write back again f x okay g prime y g that is g prime at f x plus v s into this f t minus f x is f prime x plus ut into t minus x okay and now I think we have all the all the required things for the proof so so consider t is not equal to x then we can divide by t minus x okay so let us say that t not equal to x then divide by this t minus x so what we will get is h t minus h x divided by t minus x that will be same as this g prime of at f x plus v s into f prime at x plus ut and to show that h is differentiable at x means that we should show that limit of this left hand side exists as t goes to x okay as t goes to x what happens look at the right hand side as t goes to x ut tends to 0 as t goes to x ut tends to 0 so this limit of this becomes just f prime at x what about this bracket this is independent of t okay g prime f x is a constant what about v s now here we have to use the find that when a function is differentiable it is also continuous so g is differentiable means g is g is also continuous so essentially we want to search v s goes to 0 as s goes to that is f t goes to f x that is the find that we have to use f t goes s goes to s goes to y which is same as here f t goes to f x which means f is continuous but that is true because f is differentiable at x so that is why v s goes to 0 as as s goes to y or s goes to y means s goes to f x so this tends to g prime of f x into f prime at x okay so we so we have proved that h is differentiable and its limit is nothing but g prime of f x into f prime of x okay let us now just see a couple of examples of using these theorems that is both the earlier theorem about the sums products etcetera and this chain root let me just take this example first okay f x is equal to something that we have seen earlier x sin 1 by x for x not equal to 0 and equal to let us say 0 for x equal to 0 okay now if x is not equal to 0 there is no problem that is it is x this is this function is differentiable everywhere if at all there is any problem it occurs because of this function sin 1 by x okay but again if you use the chain rule we let us assume that sin is differentiable everywhere and its derivative is cos of whatever is inside okay so so the the problem is about this function 1 by x okay then function 1 by x okay the one is differentiable everywhere that is a constant function so you take 1 by x so that will be differentiable at those points wherever x is not 0 right so this product is differentiable every at all x not equal to 0 the only point that needs to be check this what happens at x equal to 0 all right so at x equal to 0 again you think of this f t minus f x divided by t minus x so that is okay now we are looking at derivative at 0 so let us look at f t minus f 0 as divided by t minus 0 okay so that is nothing but t sin 1 by t minus f 0 is 0 divided by t okay and that is nothing but sin 1 by t and the question is whether the limit of this sin 1 by t exists as t goes to 0 okay and we already seen that this limit does not exist okay we already seen that this limit does not exist okay and so so what is the conclusion that f is not differentiable at 0 what is the conclusion f is not differentiable at 0 remember that we have already shown that f is continuous everywhere okay f is continuous everywhere okay and so we have got one more example of a function which is continuous but not differentiable okay this function is continuous at 0 but it is not differentiable at 0 now let me consider a small modification of that so now I think of f x is instead of x sin 1 by x suppose I consider x square sin 1 by x for x not equal to 0 okay and say the same this here it is set 0 for x equal to 0 okay again you will see that for x not equal to 0 there is no problem okay f is because if x is not equal to 0 you have product of the two functions and this is again composite function so it is it is differentiable and we can find a value derivative if we want by using product rule under chain rule okay if necessary we shall do it later okay so the real point is because there is what happens at x equal to 0 right let us say again we do the same thing so suppose we take this f t minus f 0 divided by t minus 0 then this will be t square sin 1 by t minus 0 divided by t minus 0 t minus 0 so what is that so this is t sin 1 by t okay t sin 1 by t and we have already seen that the limit of this as t goes to 0 exist okay and that limit is nothing but that limit is nothing but 0 so this tends to 0 as t tends to 0 which means this function is differentiable this function is differentiable at x equal to 0 and its value is value of the limit is 0 okay so so with the conclusion is so f is differentiable at x equal to 0 and what about the value and f prime 0 is 0 right f prime 0 is 0 okay so as far as x not equal to 0 is concerned we already know that f is differentiable okay so we can say that f is differentiable for all x in r okay f is f is differentiable on r okay again this is something let me explain I have not mentioned it earlier if is if f is differentiable at all points in some interval just as if f is continuous at all points in interval we say that f is continuous on a b similarly if it is differentiable at all points on some interval we said it is differentiable on that interval on that interval or on that set in this case that set turns out to be r okay so f is differentiable everywhere on r okay let us say suppose I now I want to ask a question is this f prime a continuous at 0 is f prime continuous at 0 if that is to be answered what we will have to do we will have to find what is we know that f prime at 0 is 0 so we will have to look calculate f prime x for x not equal to 0 also and take the limit of that as x goes to 0 that can be done right let us say let us say let us do this okay what is for x not equal to 0 what is f prime we will have to calculate this f prime by using two theorems proved earlier namely the product rule and the chain rule okay we can do it quickly so what is f prime x so let us it will be 2x into sin 1 by x right that is derivative of this into this okay multiplied by plus x square into derivative of this first first it is cos 1 by x and then multiplied by minus 1 by x right okay so this becomes okay this will remain as it is so 2x sin 1 by x and minus cos 1 by x right okay now if we want to know that whether f prime is continuous at x equal to 0 we will have to take the limit of this as x goes to 0 limit of this as x goes to 0 and show that that limit is same as 0 okay now what is the answer here does the limit of this exists as x goes to 0 again as far as this first term is concerned there is no problem x sin 1 by x limit of that exists as but what about the second term cos 1 by x the limit does not exist the limit does not exist so this function is differentiable everywhere but it is not continuous at x equal to 0 that is f prime is not continuous at x equal to 0 right is it clear and what is the type of discontinuity it is what we call discontinuity of the second time because the limit does not neither left hand limit nor the right hand limit will exist okay so this is the discontinuity of the second time okay fine okay we will stop with this for today we shall continue with the discussion of this different types of discontinuities etcetera in the in the next class.