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Our goal in this set of lecture notes is to provide students with a strong foundation in mathematical analysis. Such a foundation is crucial for future study of deeper topics of analysis. Students should be familiar with most of the concepts presented here after completing the calculus sequence. However, these concepts will be reinforced through rigorous proofs.
The lecture notes contain topics of real analysis usually covered in a 10-week course: the completeness axiom, sequences and convergence, continuity, and differentiation. In addition, the notes include many carefully selected exercises of various levels of difficulty. Hints and solutions to selected exercises are available in the back of the book. For each section, there is at least one exercise with hints or fully solved. For those exercises, besides the solutions, there are explanations about the process itself and examples of more general problems where the same technique may be used.
The last chapter contains additional topics. These include topological properties of the real line, generalizations of the extreme value theorem and more contemporary topics that expand on the notions of continuity or optimization. Lower and upper semicontinuity, differentiation of convex functions, and generalized differentiation of non-differentiable convex functions can be used as optional mathematical projects.
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Portland State University Library
Mathematical analysis -- Foundations, Mathematics
© 2022 Beatriz Lafferriere, Gerardo Lafferriere, and Mau Nam Nguyen
This open access textbook is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
Lafferriere, Beatriz; Lafferriere, Gerardo; and Nguyen, Mau Nam, "Introduction to Mathematical Analysis I - 3rd Edition" (2022). PDXOpen: Open Educational Resources. 43.