reference request - Need for a Analysis Book with the following requisites

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 dierentiability of real
valued functions of one variable and applications, uniform convergence, Riemann
integration, continuity and dierentiability of real valued functions of several variables, partial derivatives and mixed partial derivatives,total derivatives.

Looking forward to all of you for an answer.