As a foundation for all further results, basic algebraic structures are studied, with a particular focus on the axiomatic description of the ordering structure of the real numbers. The absolute value for real and complex numbers, metric spaces, as well as the concept of norms in vector spaces, are introduced along with elementary topological concepts. Sequences and series (particularly power series) and their convergence are studied. Limits and continuity of real and complex functions (as well as functions on metric spaces) are another topic, along with central theorems about continuous functions (Intermediate Value Theorem, Theorem on the continuous image of compact sets, Extreme Value Theorem, Uniform Continuity Theorem). Another central topic is the differentiation of functions of a real variable. This is treated in detail, with the most important differentiation rules being proven. Applications of differentiation (Rolle’s Theorem, Mean Value Theorems, Monotonicity, Maxima and Minima, Convexity, Taylor’s Formula, Taylor Series) are thoroughly examined. Elementary functions
such as polynomials, rational functions, exponential functions, general powers, logarithms, trigonometric functions, and their inverse functions are also introduced, and their properties are derived. In all the mentioned topics, emphasis is placed on logical structure, and the necessary proof methods are also treated in detail.