Basics. - Sets. - Operations on Sets. - Functions. - Operations on Functions. - Number Systems. - Countable and Uncountable Sets. - Sequences. - to Sequences. - Limsup and Liminf. - Some Special Sequences. - Series. - to Series. - Elementary Convergence Tests. - Advanced Convergence Tests. - Some Particular Series. - Operations on Series. - The Topology of the Real Line. - Open and Closed Sets. - Other Distinguished Points. - Bounded Sets. - Compact Sets. - The Cantor Set. - Connected and Disconnected Sets. - Perfect Sets. - Limits and the Continuity of Functions. - Definitions and Basic Properties. - Continuous Functions. - Topological Properties and Continuity. - Classifying Discontinuities and Monotonicity. - The Derivative. - The Concept of Derivative. - The Mean Value Theorem and Applications. - Further Results on the Theory of Differentiation. - The Integral. - The Concept of Integral. - Properties of the Riemann Integral. - Further Results on the Riemann Integral. - Advanced Results on Integration Theory. - Sequences and Series of Functions. - Partial Sums and Pointwise Convergence. - More on Uniform Convergence. - Series of Functions. - The Weierstrass Approximation Theorem. - Some Special Functions. - Power Series. - More on Power Series: Convergence Issues. - The Exponential and Trigonometric Functions. - Logarithms and Powers of Real Numbers. - The Gamma Function and Stirling's Formula. - An Introduction to Fourier Series. - Advanced Topics. - Metric Spaces. - Topology in a Metric Space. - The Baire Category Theorem. - The Ascoli-Arzela Theorem. - Differential Equations. - Picard's Existence and Uniqueness Theorem. - The Method of Characteristics. - Power Series Methods. - Fourier Analytic Methods. - Glossary of Terms from Real Variable Theory. - List of Notation. - Guide to the Literature. Language: English