I. Introduction. II.Understanding Smoothness. 2. Proving Propositions. 3. Sequences of Real Numbers. 4. BolzanoWeierstrass Results. 5. Topological Compactness. 6. Function Limits. 7. Continuity. 8. Consequences of continuity of intervals. 9. Lower Semicontinuous and Convex Functions. 10. Basic Differentiability. 11. The Properties of Derivatives. 12. Consequences of Derivatives. 13. Exponential and Logarithm Functions. 14. Extremal Theory for One Variable. 15. Differentiation in R2 and R3.16. Multivariable Extremal Theory. III. Integration and Sequences of Functions. 17. Uniform Continuity. 18. Cauchy Sequences of Real Numbers. 19. Series of Real Numbers. 20. Series in Gerenal. 21. Integration Theiry. 22. Existence of Reimann Integral Theories. 23. The Fundamental Theorem of Calculus (FTOC). 24. Convergence of sequences of functions. 25. Series of Functions and Power Series. 26.Riemann Integration: Discontinuities and Compositions. 27. Fourier Series. 28. Application. IV. Summing it All Up. 29. Summary. V. References. VI. Detailed References.;Basic Analysis I: Functions of a Real Variable is designed for students who have completed the usual calculus and ordinary differential equation sequence and a basic course in linear algebra. This is a critical course in the use of abstraction, but is just first volume in a sequence of courses which prepare students to become practicing scientists. This book is written with the aim of balancing the theory and abstraction with clear explanations and arguments, so that students who are from a variety of different areas can follow this text and use it profitably for self-study. It can also be used as a supplementary text for anyone whose work requires that they begin to assimilate more abstract mathematical concepts as part of their professional growth. Features Can be used as a traditional textbook as well as for self-study Suitable for undergraduate mathematics students, or for those in other disciplines requiring a solid grounding in abstraction Emphasises learning how to understand the consequences of assumptions using a variety of tools to provide the proofs of propositions