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I am a maths graduate .I have to appear for an entrance exam which has in its syllabus the following.

Background:

I have Read Analysis from Bartle Sherbert and Understanding Analysis:Abbott.

Requisite:

I am in need of a book which contains examples and many problems on the given topics and if possible some hints to selected problems.As I am doing self study having hints is a great advantage for me.

On surfing similar questions I found that people have mostly recommended Rudin :Principles of Mathematical Analysis.Is it suitable for self study?Also there are so many books under the topic Good books for self study in Analysis that it is difficult to select one.

Also the type of questions that came in the previous exam like :If a continuous function is injective then it is either increasing/decreasing. are also not available there.Please suggest some books for these type of questions to deal with in the exam.

Syllabus:

1.General topology: Topological spaces, continuous functions, connectedness, compactness, separation axioms, product spaces, quotient topology, complete metric spaces, uniform continuity, Baire category theorem.

2.Real analysis: Sequences and series, continuity and di erentiability of real valued functions of one variable and applications, uniform convergence, Riemann integration, continuity and di erentiability of real valued functions of several variables, partial derivatives and mixed partial derivatives,total derivatives.

Looking forward to all of you for an answer.

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  • $\begingroup$ I would recommend Introduction to Analysis by Rosenlicht: amzn.to/2fKFEdS $\endgroup$ – Math1000 Nov 12 '16 at 8:47
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Rudin is good as a reference book, but many feel that it is too difficult (and brief) for self-learning, especially if you are encountering the material for the first time.

For General Topology, I would recommend Munkres: https://www.amazon.com/Topology-2nd-James-Munkres/dp/0131816292

For Analysis (excluding measure theory): I would recommend Spivak's Calculus (https://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098918). Despite its name it is not a calculus book, it is actually an excellent analysis book.

For the remainder of the analysis stuff that is not covered in Spivak, you may want to try Apostol or Royden books.

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    $\begingroup$ I think royden's exposition of the materials listed in the OP are pretty brief, it seems to me that the book has a bent towards lebesgue integration and functional analysis (at least to some extent.) A copy can be found online here $\endgroup$ – Andres Mejia Nov 12 '16 at 9:00
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I think that Terence Tao has a lot of nice "quizzes" here.

I would actually also recommend his book Real Analysis I. There are some sample chapters in the link provided. It is full of examples, contains nice exercises and moreover, I just think it is well written and lucid.

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