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I am going through Andreescu et al.,Problems in Real Analysis: Advanced Calculus on the Real Axis and I am very impressed: the style of the book seems really modern and the material covered includes many theorems, examples, etc. that are rarely or not at all seen in other books (both theory books and problem books) and that are often taken from not well-known sources or from recent issues of math journals.

I was wondering if there is a "theory book" that resembles the one I mentioned (and roughly covers the same material)?

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I've seen the book you mention before, but it's been over a year, so I don't recall exactly the nature of the material you're seeking - rare stuff not in other books, or material covered in books like Rudin/Ross?

If you're looking for creative/nonstandard "Putnam" questions over the standard material (which I believe is the nature of the book you mention), try the three volume "Problems in Mathematical Analysis" series by Kaczor and Nowak. Lang's "Undergraduate Analysis" (with accompanying solution manual by Shakarchi) also covers some "beyond the box" material.

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  • $\begingroup$ Could you explain what kind of "beyond the box" material Lang covers? $\endgroup$ – Dal Jan 3 '15 at 0:12
  • $\begingroup$ Hi Dal, Lang has some good problems on uniform convergence, differentiating and integrating series, differentiating under the integral sign, Fourier series, ideas like Stirling's Formula and introductory topology that don't get a lot of attention in recent undergrad books introducing analysis, etc. Lang tends to have a hurried feel; Apostol is a more well-organized resource IMO. But the answer guides by Shakarchi have been, personally, indispensable in helping me cover important topics I missed in my undergrad that have been skipped in my grad courses. $\endgroup$ – Darrin Jan 4 '15 at 19:41

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