This book occupies a niche between a calculus course and a full-blown real analysis course. Its charm is that it gives very thorough and leisurely explanations, in a discursive style: You just read along about some interesting properties of the real numbers and then find them codified as a definition or theorem, rather than being confronted with a mass of definitions, theorems, and proofs.
The book assumes the student has already been through calculus, but without the proofs. It presents most of the important ideas of real analysis without requiring any great conceptual leaps from a calculus course. It is very carefully positioned to lie between a non-rigorous calculus course and a real analysis course such as might be taught from Rudin's Principles of Mathematical Analysis or Apostol's Mathematical Analysis.
The book is mathematically not very ambitious, and at first glance it may look like there's not much here. I think the book should be viewed as a text for a bridge or transition course that happens to be about analysis, rather than an analysis course per se.
Certainly it has much less material than would normally be found in an analysis text. There's not much topology, no construction of the real numbers (there's a brief sketch of Dedekind cuts), no measure theory or Lebesgue integral, and no function spaces. The book depends almost totally on completeness of the reals for its proofs, although the Bolzano-Weierstrass theorem is introduced to back up uniform continuity, which is needed for integrability.
There are several Very Good Features:
Lots of counterexamples. Most calculus books get the proof of the chain rule wrong, and Ross not only gives a correct proof but gives an example where the common mis-proof fails. There are also examples of failures of L'Hospital's rule and of non-integrable functions.
Introduces limits of sequences first, and only then goes on to continuity
Lengthy discussion of the Riemann-Stieltjes integral, which is very handy in mixed discrete-continuous problems and number theory and which most texts don't cover at all
The book has a number of optional sections, which tend to be dead ends in this book but are interesting in themselves and in a more advanced course would have many consequences. Most of the topology material is in this category. There's a complete proof of the Weierstrass approximation theorem (using Bernstein polynomials), which doesn't go anywhere in this book but is certainly a startling result.
Allen Stenger is a math hobbyist, library propagandist, and retired computer programmer. He volunteers in his spare time at MathNerds.com, a math help site that fosters inquiry learning. His mathematical interests are number theory and classical analysis.