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# Study BSc - Real Analysis

UNIT I 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. UNIT II 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. Unit III 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. Unit IV 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. Unit V 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.

## BSc Subjects

 Applied Electronics Atomic Physics and Nuclear Physics Classical Algebra and Calculus CLIENT SERVER COMPUTING Climatology and Oceanography Complex Analysis DATA COMMUNICATION AND NETWORKS DATA STRUCTURES AND C PROGRAMMING Differential Equation Lap lace Transforms DIGITAL FUNDAMENTALS AND ARCHITECTURE Economic geography Geography of India Geomorphology Heat Thermodynamics Mechanics Properties of Matter and Sound Human geography INTERNET and JAVA PROGRAMMING MATHEMATICAL FOUNDATION FOR COMPUTER Mathematical Physics Mechanics Modern Algebra OBJECT ORIENTED PROGRAMMING WITH C++ Optics Electricity and Magnetism Principles of cartography Quantum Mechanics and Relativity Real Analysis SOFTWARE ENGINEERING Solid State Physics Statistics for geographers Statistics for Mathematics SYSTEM SOFTWARE AND OPERATING SYSTEM Trigonometry Vector Calculus and Analytical Geometry VISUAL PROGRAMMING WEB TECHNOLOGY Business Economics Business Management Business Organisation and Office Management Business Statistics Commercial Law and Company Law Company law and secretarial practice I Company law and Secretarial practice II Corporate Accounting Cost and Management Accounting Executive Communication Financial Accounting Industrial law Life Office Management Managerial Economics Mathematics for Business Principles of Accountancy Principles of Life Insurance Principles of Management Principles of Marketing Priniciples of Reinsurance Statistics

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