Real Analysis: A Comprehensive Guide

This subsection begins by acknowledging the student's prior calculus knowledge of the real number system but emphasizes that all its properties derive from a few fundamental axioms. While the full development of the real numbers from these axioms isn't undertaken, the text stresses their importance as a foundation for real analysis. The focus shifts to the property of completeness, presented as something likely new to the student, setting the stage for a rigorous treatment of calculus. The text states that a set on which two operations are defined, having specific properties, is called a field. It then provides examples of fields (real numbers and rational numbers). This introduction emphasizes the importance of a solid understanding of the axioms that underpin real analysis.

This subsection introduces the Completeness Axiom, a crucial element for rigorous calculus. The text highlights the distinction between defining a mathematical object and proving its existence, using the example of trying to define the smallest positive real number. The concept of the supremum of a set is discussed, noting that the empty set doesn't have a supremum. The author emphasizes that properties (A) - (H), while significant, don't guarantee the existence of a supremum for every nonempty set bounded above. Therefore, the Completeness Axiom (I) is introduced: 'If a nonempty set of real numbers is bounded above, then it has a supremum'. This axiom, central to the rigorous development of calculus, is presented as a defining characteristic of the real number system as a complete ordered field. The uniqueness of the real number system, up to symbolic differences, as the only complete ordered field is also mentioned.

This subsection extends the real number system by introducing two fictitious points, +∞ and -∞, to handle unbounded sets. It defines the order relationships between these points and any real number x. The concept of sets unbounded above or below is introduced, clarifying the necessity of these additions for convenient mathematical representation. The section concludes with an exercise focusing on finding formulas for the supremum and infimum of a transformed set (T) based on the supremum and infimum of the original set (S).

This section focuses on the rigorous definition and properties of limits of functions of a single variable. It delves into the epsilon-delta definition of a limit, providing a formal framework for understanding the concept of a function approaching a specific value as its input approaches a certain point. The discussion includes limits at infinity and infinite limits, expanding the scope beyond finite values. The text also highlights the relationship between different forms of limit definitions (e.g., using absolute values versus inequalities) and their equivalence. This formal treatment lays the groundwork for later topics such as continuity and differentiability, emphasizing precise mathematical language and notation.

Building upon the foundation of limits, this section defines continuity and explores its various aspects. It covers removable discontinuities, showing how some discontinuities can be 'fixed' by redefining the function at a single point. The concept of composite functions and their continuity is addressed. The section also discusses bounded functions, functions whose range is limited within a certain interval. The intermediate value theorem is presented, highlighting an important property of continuous functions. Finally, uniform continuity, a stronger form of continuity, and additional properties of monotonic functions are examined. This section establishes a deep understanding of continuous functions, using precise language and mathematical rigor.

This section delves into the theoretical properties of differentiable functions, assuming the student is familiar with basic differentiation rules from calculus. The derivative is defined, and its geometric interpretation as the slope of the tangent line to a curve is given. Topics include the interchange of differentiation and arithmetic operations, the chain rule for differentiating composite functions, and one-sided derivatives. The section also examines extreme values of differentiable functions, Rolle's theorem, the intermediate value theorem for derivatives, and the mean value theorem, emphasizing the rigorous theoretical understanding of these concepts. The text emphasizes understanding principles over recalculating familiar formulas.

This section introduces the Riemann integral in the standard way, using Riemann sums. The integral is defined as the limit of these sums as the partition of the interval becomes increasingly fine. The concept of upper and lower integrals is also introduced, providing a framework for examining the existence of the Riemann integral for a given function. The text uses the intuitive idea of the area under a curve as motivation, but emphasizes that this is not a rigorous definition, rather the area is defined to be the integral's value (if it exists). This section lays the mathematical foundation for the subsequent exploration of the Riemann integral's properties and applications, highlighting the rigorous definition of the integral using Riemann sums. The focus is on bounded functions given a prior theorem.

This section addresses the conditions necessary and sufficient for the existence of the Riemann integral. The approach leverages upper and lower sums and integrals to analyze a function's integrability. The text shows that continuous functions and bounded monotonic functions are always Riemann integrable. This section provides the criteria for determining whether the Riemann integral exists for a given function, offering both necessary and sufficient conditions. It focuses on the relationship between upper and lower sums and integrals, providing a practical means to assess integrability.

This section focuses on the properties of the Riemann integral, proving that the sum and product of Riemann integrable functions are also integrable. It shows that the absolute value of an integrable function is also integrable. Other key properties are discussed, including the first mean value theorem for integrals, the concept of antiderivatives, the fundamental theorem of calculus, techniques like change of variables and integration by parts, and the second mean value theorem for integrals. This section establishes the key algebraic and analytic properties of the Riemann integral, providing tools for evaluating and manipulating integrals. It highlights theorems that are crucial for practical applications of integration.

This section provides a comprehensive discussion of improper integrals, which are integrals where either the interval of integration is unbounded or the function is unbounded at one or both endpoints. Concepts like absolute and conditional convergence of improper integrals are defined and explored, along with Dirichlet's test for convergence. The section also covers change of variables in improper integrals. The author states that the treatment of improper integrals in this book is more detailed than in most comparable textbooks. This section expands the scope of integration to include cases not covered by the standard definition, providing rigorous analysis for unbounded functions or infinite intervals.

This section presents Lebesgue's criterion for the Riemann integrability of a bounded function. It defines the concept of a set with Lebesgue measure zero and states that a bounded function is Riemann integrable on an interval if and only if the set of its discontinuities has Lebesgue measure zero. This is presented as a more practical criterion for determining integrability than those requiring computations of upper and lower sums. The section concludes by noting that this section can be omitted without impacting the understanding of subsequent sections. This section presents a powerful, advanced criterion for integrability, relying on the concept of Lebesgue measure and offering an alternative to the more computationally intensive methods.

This section begins the study of sequences and series by focusing on sequences of constant terms. The concepts of limit inferior and limit superior are introduced, providing tools for analyzing the convergence behavior of sequences. The author explains that the choice to include limit inferior and limit superior in the development allows for greater flexibility and generality in the later study of infinite series. This foundational section establishes essential terminology and concepts, focusing on the convergence properties of sequences. The inclusion of limit inferior and limit superior is highlighted for its impact on subsequent topics.

This section explores the relationship between sequences and the concepts of continuity and differentiability. It demonstrates how sequences can be utilized as tools to investigate these properties of functions. This provides an alternative perspective to the epsilon-delta definitions, offering a different lens for understanding continuity and differentiability. The section serves as a bridge, linking the theory of sequences to the previously covered concepts of continuity and differentiability, demonstrating a powerful connection between these crucial areas of analysis.

This section shifts the focus from sequences to infinite series of constant terms. It introduces concepts of convergence and divergence, laying the groundwork for determining whether an infinite sum converges to a finite limit or diverges. Cauchy's convergence criterion is presented, providing a powerful test for convergence. For series with positive terms, various convergence tests, such as the comparison test, the integral test, the ratio test, and Raabe's test are introduced and explained. The section also tackles the more general case, covering absolute and conditional convergence, Dirichlet's test, and the behavior of series under rearrangement of terms and multiplication of one series by another. This section provides a thorough exploration of infinite series, equipped with various tests for convergence and divergence.

This section serves as a review of essential concepts from linear algebra, acknowledging that these concepts are integral to the differential calculus of vector-valued functions. The review includes matrices, determinants, and linear transformations, which form the groundwork for understanding the subsequent material. The concepts are not developed from first principles, indicating an assumption of prior knowledge. This section bridges the gap between linear algebra and the subsequent discussion of vector-valued functions, ensuring a solid foundation for understanding the advanced calculus concepts.

This section defines the differential of a vector-valued function as a linear transformation, extending the concept of the derivative from single-variable calculus to the multivariable setting. The chain rule is discussed within this framework, showing how it applies to the composition of vector-valued functions. The emphasis is on understanding the differential as a linear transformation, rather than just a vector. This section connects the familiar concepts of derivatives and chain rules to the multivariable context, highlighting the power of a linear transformation approach.

This section covers the inverse function theorem, a key result in multivariable calculus. The theorem is explained, introducing the concept of branches of an inverse function in the context of vector-valued functions. The section likely focuses on the theoretical aspects of the inverse function theorem and its implications for understanding the local invertibility of functions, extending the single variable theorem to vector valued functions. This section addresses a significant theorem in multivariable calculus, introducing the concept of branches of inverse functions to extend the single-variable theorem's applicability.

The implicit function theorem is the subject of this section. The theorem is motivated through the lens of linear transformations before being stated and proved in its general form. This indicates a gradual approach from simpler cases to the more complex general theorem. The implicit function theorem is a powerful tool for analyzing functions defined implicitly, enabling a better understanding of relationships between variables that are not explicitly defined as functions of one another. This is particularly useful in contexts where it is difficult or impossible to solve explicitly for one variable in terms of the others.

This section introduces the integral calculus of real-valued functions of several variables, extending the integration concepts from single-variable calculus to multiple dimensions. Multiple integrals are defined initially over rectangular parallelepipeds, and the definition is then extended to more general sets. The text specifically addresses the case where the function's discontinuities form a set of Jordan content zero. Evaluation using iterated integrals is discussed, along with the concept of Jordan measurability and change of variables. The section acknowledges the complexity of the change of variables proof, noting its unavoidable intricacy. This section expands integration to the multivariable setting, covering multiple integrals, iterated integrals, and the crucial change of variables technique. It requires a thorough understanding of the prior sections on linear algebra and vector-valued functions.