I Derivatives.- § 1. First Derivative.- § 2. The Mean Value Theorem.- § 3. Derivatives of Higher Order.- § 4. Convex Functions of a Real Variable.- Exercises on §1.- Exercises on §2.- Exercises on §3.- Exercises on §4.- II Primitives and Integrals.- § 1. Primitives and Integrals.- § 2. Integrals Over Non-Compact Intervals.- § 3. Derivatives and Integrals of Functions Depending on a Parameter.- Exercises on §1.- Exercises on §2.- Exercises on §3.- III Elementary Functions.- § 1. Derivatives of the Exponential and Circular Functions.- § 2. Expansions of the Exponential and Circular Functions, and of the Functions Associated with Them.- Exercises on §1.- Exercises on §2.- Historical Note (Chapters I-II-III).- IV Differential Equations.- § 1. Existence Theorems.- § 2. Linear Differential Equations.- Exercises on §1.- Exercises on §2.- Historical Note.- V Local Study of Functions.- § 1. Comparison of Functions on a Filtered Set.- § 2. Asymptotic Expansions.- § 3. Asymptotic Expansions of Functions of a Real Variable.- § 4. Application to Series with Positive Terms.- Exercises on §1.- Exercises on §3.- Exercises on §4.- Exercises on Appendix.- VI Generalized Taylor Expansions. Euler-Maclaurin Summation Formula.- § 1. Generalized Taylor Expansions.- § 2. Eulerian Expansions of the Trigonometric Functions and Bernoulli Numbers.- § 3. Bounds for the Remainder in the Euler-Maclaurin Summation Formula.- Exercises on §1.- Exercises on §2.- Exercises on §3.- Historical Note (Chapters V and VI).- VII The Gamma Function.- § 1. The Gamma Function in the Real Domain.- § 2. The Gamma Function in the Complex Domain.- Exercises on §1.- Exercises on §2.- Historical Note.- Index of Notation.