Syllabus
Lectures
Number systems and their properties. Affinely extended real number line R* and its arithmetics. Intervals.
Supremum and infimum for R and R*.
Mapping (general function). Input and output of function, doman and codomain, image and inverse of set. Composition, restriction of function. Inverse fuction, its properties and use.
Function (with codomain in number set). Extrema, supremum and infimum of function. Operations on functions. Injective function.
Boundness in ordered sets and in metric spaces. Case of real numbers.
Monotonic and strictly monotonic functions. Local monotonicity and local extrema. Relation between monotonicity and local monotonicity in Q and in R. Inverse of strictly monotonic function.
Convexity and concavity of function: two definitions, geometric interpretations and equivalence. Inverse of convex and concave function.
Parity and periodicity of function. Properties of even and odd functions, decomposition to even part and odd part. Set of periods, fundamental period. Relation between parity and periodicity of operands and result for arithmetic operations and composition.
Basic real functions: constant, powers, roots, exponential functions, logarithms, trigonometric and inverse trigonometric functions. Properties and formulas. Sign function, indicator function, Dirichlet function.
Elementary functions. Continuity of elementary functions and its consequences. Elementarity of basic functions. Examples of non-elementary functions.
Seminar
Solving of inequalities in R, determining domain of elementary functions, linear transformations of graphs of functions, determining inverse of elementary functions.
Annotation
Basic theory of real functions, elementary functions and methods for solving standard problems connected with them.