The Real and Complex number systems the field axioms, the order anixms, integers,
the unique Factorization thorium for intcgcrs, Rational numbers, Irrational numbers --- Upper
bounds , Maximum Elements, least upper bound, the completeness axiom, some properties for
the sypremum, properties of the integers deduced from the completeness anxiom- The
Archimedian property of the real number system Rational numbers with finite decimal
representation of real numbers absolute values and the triangle inequality – the Cauchy-
Sohewarz,inequality-plus and minus infinity and the extended real number system.
Basic notions of asset theory. Notations –ordered pairs – Cartesian product of two sets,
Relations and functions further terminology concerning functions one one functions and inverse
composite functions- sequences-similar sets – countable and uncountable sets- uncountability of
the real number system-set algerbra-countable of collection of countable sets.
Elements of point set topology: Euclidean space R". The strcture of open Sets in R" –
closed sets and adherent points- The Bolzno – Weierstrass theorem – the Cantor intersection
theorem. Covering Lindelof covering theorem the Heine Borel covering Compactness in R"
Metric Spaces – Point set topology in metric spaces – compact subsets of a metric space –
Boundary of a set.
Convergent sequences in a metric space- Cauchy sequences – complete metric Spaces.
Limit of a function Continuous functions composite functions. Continuous complex valued
functions. Examples of continuous functions - continuity and inverse images of open or closed
sets – functions continuous on compact sets – Topological mappings – Bolzano’s theorem.
Connectedness – components of metric space – Uniform continuity and compact setsfixed
point theorem for contractions – monotonic functions.Definition of derivative-Derivative
and continuity- Algebra of derivatives- the chain rule – one sided derivatives and infinitives
derivatives- functions with non-zero derivatives-zero derivatives and local extrema – Roll’s
theorem – the mean value theorem for derivatives- Taylor’s formula with remainder.
Properties of monotonic functions- functions of bounded variation – total Variation –
additive properties of total variation on (a,x) as a function of x- functions of bounded variation
expressed as the difference of increasing functions- continuous functions of bounded variation.
The Riemann – Stieltjes integral : Introduction – Notation – The definition of Riemann stieltjes
integral Reduction to a Riemann integral.