Continuum Mathematics

1. Basic concepts of set theory. Relations on sets. Mappings between sets. Basics of
enumerative combinatorics.
2. The sets N,Z,Q,R of natural numbers, relative integers, rational and real numbers, with their
algebraic structures and order relations. Completeness of R. The spaces R^n (n=1,2,3, ).
3. A sketch of the abstract notion of vector space. Linear mappings between vector spaces and
matrices. Linear equations.
4. Some generalities on real functions from a subset of R to R. The notions of limit and
continuity for such functions. Some elementary functions: polynomials, exponential,
logarithm, trigonometric functions.
5. Real sequences and their limits. Real series.
6. The derivative of a real function of one real variable . A discussion on the geometrical
meaning and on the applications of the derivative. Derivatives of elementary functions.
Basic facts about derivable functions.
7. Higher order derivatives. Taylor's formula. Use of Taylor's formula in the computation of
limits. Taylor's series.
8. How to use the derivatives to determine the maximum and minimum points of a real
function of one real variable, as well as the intervals where the function is increasing,
decreasing, convex or concave.
9. The theory of Riemann's integral for real functions of one real variable. Geometrical
meaning of Riemann's integral. The ''fundamental theorem of calculus''. Basic integration
rules.
10. The field C of complex numbers. Modulus, argument and trigonometric representation of a
complex number. A sketch of the notions of limit, derivative and integral for complex
valued functions. Complex sequences and series. The exponential function in the complex
field; Euler's formula. Roots of complex numbers. The fundamental theorem of algebra.