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Suggest books,which are full of limit and continuity solved problems,in undergraduate level.(real analysis)

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Walter Rudin's mathematical analysis Also Thomas Calculus

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    Rudin's Principles doesn't come with solved problems to exercises in the book. – 2017-02-05
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    There's a suggested solution manual, although not so encouraged. – 2017-02-06
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For introductory real analysis Numbers and Functions: Steps into Analysis by R. P. Burn.

For real and complex analysis from the metric space point of view A Second Course In Mathematical Analysis by J. C. Burkill.

Both books come with solutions to almost every exercise.

Disclaimer: I am in the process of reading Burn's book and recommend Burkill's book based on Burn's recommendation of it in the bibliography.