This article needs additional citations for. Unsourced material may be challenged and removed. ( March 2013) are of utmost importance in, functions and applications.
However, not all are continuous. If a function is not continuous at a point in its, one says that it has a discontinuity there.
The set of all points of discontinuity of a function may be a, a, or even the entire domain of the function. This article describes the classification of discontinuities in the simplest case of functions of a single variable taking real values.
Mathematical Analysis has 170 ratings and 8 reviews. Offers an outline of the essential properties of rational numbers using Dedekind's cut, and establis.
The of a function at a point quantifies these discontinuities as follows:. in a removable discontinuity, the distance that the value of the function is off by is the oscillation;. in a jump discontinuity, the size of the jump is the oscillation (assuming that the value at the point lies between these limits from the two sides);.
in an essential discontinuity, oscillation measures the failure of a limit to exist. A special case is if the function diverges to infinity or minus infinity, in which case the oscillation is not defined (in the extended real numbers, this is a removable discontinuity).
Disclaimer: All the things mentioned below is based on my experience. It is not the only strategy. I humbly believe in principle of Anekantavada. Dear all, I have written 4 mains with maths and secured variety of marks in them. I got 238(122+116) in 2014, 195 (122+73) in 2015, 288 (145+143) in 2016 and 310 (147+163) in 2017. I have interacted with numerous aspirants who took mathematics in these 4 years.
I would like share various aspects that I have observed in these years. I did no coaching for maths. Neither did I join any test series. Choosing Mathematics as an optional Many aspirants take mathematics as an optional and after exhausting 2-3 attempts with maths, realized that maths was not the right optional for them.
Changing optional is the most unfortunate thing that can happen to an aspirant. Therefore decision that whether to take maths or not needs a lot of self-assessment and introspection.
Aspirants many times are attracted by the high marks obtained by toppers every year, which gives illusion that maths is a scoring subject. People who are the top scorers in maths (lets say top 20-25 people) will get 280-300+ marks. Rest are bound to get less marks because of scaling.
It is just the beauty of scaling method which leads to wide variety of score. I got 73 in paper II in 2015.
Since I was in lower percentile of candidates, scaling decreased my marks by a big factor. There is no time for learning basic mathematics while reading about the optional. One should be comfortable with basic topics. He/she should improvise on easy topics and learn the topics which are difficult by doing a lot of practice. Usually Btech or BSc/MSc maths people take mathematics as optional.
Before taking mathematics one should be comfortable with easy topics such as linear algebra, Calculas, Ordinary and Partial Differential Equations and Complex analysis. If one is not comfortable with these topics then he/she should think twice before talking mathematics.
Scoring good marks in maths requires competing with the aspirants who have some natural skill in maths, which even includes competing with finest IIT minds as well. There is no time for building base. Choose mathematics only if you have strong liking and interest in mathematics which includes possessing some inherent skill in maths.
Students who have performed well in their Btech/Bsc in maths courses, can infer that they have some level of comfortableness in mathematics. Coaching or no coaching Ideally a person who is good in mathematics, will not feel any need for coaching. Unlike humanities optional where many aspirants take optional without having any background of it. Science optional (especially Maths and engineering optional) is taken by people who have some background for it. So there is not much case for taking coaching so as to get some acquaintance with maths. Except for some difficult topics like Abstract Algebra, fluid mechanics there is no need for any coaching guidance. Usually all recommended books contain solved examples which are more than sufficient for teaching a new topic.
![Analysis Analysis](http://d29qn7q9z0j1p6.cloudfront.net/content/roypta/371/2002/20120417/F2.large.jpg)
But still if one thinks of going to coaching, I would advise to do it for selected topics and not for all. Joining Test series? As per present trend, UPSC maths paper level is quite tough which is not matched by any test series in the market. I think doing difficult problems from recommended books in a time bound manner is more fruitful than joining test series available presently in the market. Recommended book list and Strategy Paper 1.
Linear Alegbra: this is the one of the easiest portion in paper 1. The key lies in finding the correct answer in minimum time.
Linear Algebra book and Matrices Book by Krishna series is sufficient. One can read schaum series as well. Please do last 15 years UPSC question from this chapter for improvisation. Calculus: Questions from this chapter are also straightforward. Books needed are: Shanti Narayan – Course on Mathematical Analysis (S. Chand), mathematical analysis by S.C.
Malik and Savita Arora. For covering asymptotes and curve tracing, differential calculus by shanti Narayan needs to be studied. For practicing definite integral question (which is coming as 10 marker), Series Integral Calculus by A.R. Vashishtha (Krishna Series) can be studied. Analytical Geometry: Analytical geometry by shanti Narayan (S Chand) or Series Analytical Geometry by A R Vasishtha(Krishna Series). Krishna series book contains more solved examples so I personally prefer it.
Questions pertaining to conicoid are not easy, so for some selective good questions, how to start the question needs to be remembered. Ordinary Differential Equation: M.D.
Raisinghania (S. For Laplace, selective chapter from advance differential equation by M D raisinghania needs to be studied. Focus should be on solving question without doing ZERO calculation mistakes. Vector Analysis: Vector Analysis by Krishna series. Solve at least last 15 year question from UPSC paper. Statics: Krishna Series.
Dynamics: Krishna Series. Solve relevant chapters from Krishna series for statics and dynamics. Mostly question is coming from these books only. If a new question comes in exam which is not easy, it becomes difficult to solve in that pressure situation.
For friction: Read Golden Statics By N.P Bali. It is there on google books. Paper 2. Modern Alegbra: Group theory by R Kumar and Ring theory by R Kumar( Vardhaman Publications). Question in past 2 years have become very tricky. They are asking proof of theorems.
So approach should be to cover basics first and get comfortable level in it and then go for learning higher stuff. This portion creates a lot of challenge for majority of students as they read modern alebra for first time in their student life. Therefore ample time and lot of revisions are needed. Real Analysis: Raisinghania – Elements of Real Analysis (S.
Chand) and Mathematical Analysis by S.C. Malik and Savita Arora. Since past 2 years, there has been some very unconventional questions from this portion.
Some question in my opinion are meant to be bouncer, so nothing much can be done on that front. Therefore the best approach is to ensure that one should be able to solve easy and moderate level questions. Please do past 15 year papers from this portion. Complex Analysis: complex analysis by Krishna series.
This portion is quite straight forward. Linear Programming: Linear Programming by R.K Gupta. One of easiest topic. But ensure by adequate practice and diligence that there is no mistake in finding answer. Partial Differential Equations– M.D. Raisinghania + internet searching for practicing boundary value problem.
I will also upload some material on this. Numerical analysis: Jain & Iyengar book or any book which is there in state engineering college mathematics course.
I will upload a pdf also soon. Fluid Mechanics: M D Raisinghania. This portion is very challenging. Best bet is to memorise and practice examples form this book.
If some tough question comes in exam which is not from this book, it becomes quite unapproachable. Mechanics: Rigid Dynamics (Volume I & Volume II). It has similar case as that of Fluid mechanics. General tips. Try to solve past 20 years question papers. If time doesn’t permits, atleast solve papers for the portion that I have mentioned. Revision is the must thing.
Even if you cover some tough topic and don’t revise it enough. Touch will be lost, and all the hard done earlier will go in vain. So revise, revise and revise. My typical revision schedule after prelims was like that. Fluid mechanics, mechanics: 4 times. Statics, dynamics, modern algebra: 3 times.
Rest topics: 2 times. Focus and practice a lot on finding the correct answer.
Silly mistakes can cost you dearly. As it did in my case. Revise all formulas once in every 2 weeks. Paper 2 in 2016 and 2017 came very tough. I think this will become a norm. Please practice more for this paper. Comments are welcome ?.
Hi sir Feeling happy to talk with you I am from mathematics optional as we know we have keep revising and practicing the standard source. But how far it is important to make self notes out of these books,should we do this or avoid this Also it’s is important to to practice previous year question paper and how do you define role of test series and if yes which one? Next question related to theorem, Which portion of syllabus we need to prepare them and how,is there any special thing would you like to mentioned so as I can learn them better and most important how to remember them for long period? Very motivational reading your inspirational story. Thanks for your strategy of maths. I have written mains 2 times in 2016 (maths 270) and 2017 (maths 295). Just want to ask you about a very few specific things: 1.
Hydrodynamics – I find most topics comfortable after 2-3 times of revision. But (how) did you fully tackle chapter3 (RaiSinghania) of Euler inviscid flow( which has almost every question of very hard difficulty )? Or did you leave this chapter? Did you cover entire portion of statics and dymanics of all 11-12 chapters?
And to what extent?As some questions towards the end are very hard. 3.Did you cover proofs on complex analysis? Recent trend is there are some random proofs being asked from here.
To what extent were you thorough with proofs from modern algebra of important theorems? I am very good at problem solving(past year questions) and some proofs in this topic.
But last 2 years questions asked have been from FUNDAMENTAL theorems which are tough to prove. Did you cover Sylow groups? I could solve questions from 2000 to 2015 easily. I feel little scared as questions asked in last 2 years from this topic have been big bouncers and am wondering to what extent to prepare proofs of difficult theorems as they take hell lot of time and are difficult to recollect in exam.
5.Paper 2 Rigid dynamics- I have done Langrangian/ hamiltonain dynamics and DAlembert principle and Moment of Inertia. Did you prepare something more like small osciallations( I left this topic as questions are very long and have insane amount of arithmetic calculation) and rigid bodies dynamics? Thanks for reading this huge block of text.
Indeed it is huge block of text ? 1. I did selected problem from Euler inviscid chapter, based on my comfortableness. I covered the chapters based on the syllabus, not all chapters are required. I left tough problem at the fag end of chapter.
I left the proofs of complex analysis, it becomes very difficult to revise before the exam. If one is not able to recall proofs in exam, there is not point in covering them now. I did some easy proofs given in the R Kumar book, remembering tough & long proofs become difficult, so I left the difficult one.
Sylow groups in not in the course. I covered small oscillations, they are easy as per me.one question came in 2013 from that topic, Rigid body dynamics, i covered some easy questions from krishna series book. In general, I have covered very less no of proofs. I did silly mistakes of around 40 marks (20+20) which could have taken my score to atleast 340. So focus more on efficiency. Hi Utsav, More than any other post on your blog, I liked your journey to UPSC the most!
Particularly the last quote of Gandhi, you truly are inspiring. I have read 10’s of interviews/strategies so far, many of those with Maths optionals. But couldn’t find answer to my one common question. I request you to please take time to address this part.
1) Most of the toppers with Maths Optional toppers follow a similar set of books, but which particular chapters / sections in those books are relevant to the exam were never discussed 2) This is my first attempt and I’m having a really tough time figuring out where to stop in a particular subject (like Analytical Geometry, Real Analysis, Dynamics etc) It would be very very helpful if you could take out time to mention the relevant portion in these subjects especially for not so straightforward subjects. Thanks a ton for your help and contribution to others successes!