Introduction to Analysis by Maxwell Rosenlicht is another bargain from Dover Publications. I used this inexpensive mathematics reprint to help fill in gaps in my background before tackling more advanced mathematics. I found the first 150 pages to be challenging, but manageable. I had less success with the last 100 pages.
My college work was limited to applied mathematics, but in recent years I have developed some familiarity with metric spaces, topology, and analysis. (I previously reviewed Metric Spaces by Victor Bryant and Introduction to Topology by Bert Mendelson.)
Throughout his text Rosenlicht emphasizes how the same idea or theorem can be formulated in various ways. I found his approach to be quite helpful in clarifying more abstract representations of key ideas.
The first two chapters review set theory and the real number system and should be familiar to many readers. However, Chapter 3 (Metric Spaces) and 4 (Continuous Functions) are critical and require substantially more effort. My pace slowed dramatically.
For the reader new to metric spaces, Chapter 3 will likely be challenging, although metric space concepts are not really that difficult, just unfamiliar.
Rosenlicht demonstrates how statements concerning the open subsets of a metric space can be translated into statements concerning closed subsets, or alternatively into ones concerning sequences of points and their limits. Rosenlicht closes Chapter 3 with definitions and discussions of Cauchy sequences, completeness, compactness, and connectedness.
Rosenlicht begins Chapter 4 by illustrating that the familiar epsilon-delta definition of continuity of functions can be reformulated using the metric space open ball concept, or by using open subsets in metric spaces. He further explores the interdependence of theorems about continuity, limits, and convergent sequences. Chapter 4 concludes with discussions on continuous functions on a compact metric space and on continuous sequences of functions (analogous to sequences of points).
In chapters 5 (Differentiation) and 6 (Riemann Integration) we discuss the fundamental ideas of calculus using concepts and theorems introduced in the previous chapters. At this point I revisited a favorite calculus book by Salas, Hille, and Etgen. I was pleased to find that I now had greater insight into more advanced topics. Rosenlicht was indeed helping me.
Nonetheless, I had substantial difficulty with the longer and more complex proofs common in the remaining 100 pages, chapters titled Interchange of Limit Operations, the Method of Successive Approximations, Partial Differentiation, and Multiple Integrals. I again visited other textbooks, but this time looking for help with power series, the fixed point theorem, and the implicit function theorem. Although familiar with partial differentiation and multiple integration, I only skimmed the final chapters. I hope to return to Rosenlicht later after exploring another text on analysis.
I recommend Introduction to Analysis, especially for students looking for a review of analysis. This Dover reprint is a good buy, even if like me, you find the later chapters to be rather difficult.