Sets, relations, functions.

Axioms for the real numbers system R: the field axioms, order axioms, the completeness axiom (formulated as every nonempty set of real numbers which has an upper bound, has a least upper bound). Consequences of completeness : Dedekind's theorem and its equivalence to the completeness, R is Archimedian ordered, the set Q of rationals is dense in R, the nested intervals property of R, the existence of n-th roots for positive real numbers. Rational powers and their properties. Decimal representation of real numbers. The extended real numbers system R*. Elementary topological properties of sets of real numbers : neighborhoods, open sets, closed sets, closure and accumulation points, Bolzano-Weierstrass theorem, compact sets, Heine-Borel's theorem.

Real functions of a real variable. Operations with functions. Bounded and unbounded functions. Monotone functions.

General concepts : bounded sequences, monotone sequences, subsequences, operations with sequences. Convergence of a sequence, uniqueness of the limit. Properties of convergent sequences, operations with limits. Convergence in R*, properties. The limit of a monotone sequence. The number e, approximations for e, its irrationality. Cantor's theorem. Cauchy sequences. Cauchy's criterion for convergence. Limit points of a sequence, inferior and superior limits. The exponential and logarithmic functions, definition and their properties.

Definition (Cauchy) of the limit of a function at a point. Cauchy - Bolzano's criterion. Heine's definition for limits and its equivalence to Cauchy's definition. One sided limits. The limit of monotone functions. Limit properties and operations with limits of functions. Limit at a point of : rational functions, exponential and logarithmic functions, trigonometric functions. Inferior and superior limits of a function at a point.

Continuity of a function at a point, on a set. Discontinuity points : classification, the set of discontinuity points of first kind. The discontinuity points of a monotone function. Functions with Darboux's property , their discontinuity . Local properties of continuous functions, operations with continuous functions. Global properties of continuous functions. Bolzano's theorem , Weierstrass theorem , uniform continuity and Cantor's theorem. Continuous extension of a function. Continuity of inverse functions.

Differentiability of a function at a point, the derivative. The differential of a function. One sided derivative. Operations with differentiable functions, the derivative of a composite function. Inverse functions - their derivatives. An example of a continuous function which is not differentiable at any point. Local extrema of a function and Fermat's theorem . Rolle's theorem, Darboux's theorem, consequences. The mean-value theorem of Cauchy. Lagrange's mean-value theorem and its consequences. Taylor's formula, application . Convex function. L'Hospital's rule.