Basics of mathematical logic, quantifiers; natural numbers, Peano axioms, principle of mathematical induction; rational numbers, real numbers, completeness axiom, abstract characterisation of real numbers, algebraic numbers, complex numbers, other number systems; elements of naive set theory, cardinal numbers, countable sets, continuum; sequences, limit of a sequence and its properties, Bolzano and Weierstrass theorems, Cauchy sequences; series, convergence tests; continuity, local and global properties of continuous functions, uniform continuity; metric spaces, complete metric spaces, open, closed and compact sets; differentiation, differentiation theorems, the mean value theorem and its applications, Taylor formula; the Riemann integral and its properties, fundamental theorem of calculus, integrable functions, the Riemann-Stieltjes integral; sequences and series of functions, power series, radius and interval of convergence, analytic functions, uniform convergence, differentiation and integration of power series; introduction to functional analysis: linear spaces, normed spaces, complete normed spaces, inner product spaces.