MA20012: Real Analysis

MA20012
Course name	Real Analysis
Offered by	Mathematics
Credits	4
L-T-P	3-1-0
Previous Year Grade Distribution
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Semester	Autumn

Syllabus

Syllabus mentioned in ERP

Prerequisite: voidDedekinds definition of real numbers, field and order axioms, countable and uncountable sets, supremum and infimum of sets of real numbers, bounds and limit points of a set, Bolzano-Weierstrass theorem, open and closed sets. Limit inferior, limit superior and limit of sequence, bounded and monotonic sequences, Cauchy sequence and Cauchys general principle of convergence, product and quotient of limits, Cantors theorem on nested interval and its applications. Compact sets, Heine-Borel theorem. Limit, limit superior, limit inferior of real functions, limit theorems. Continuity and uniform continuity of real functions, properties of continuous functions, continuity and compactness. Differentiability of real functions, Taylors and Maclaurins theorems. Riemann integration, conditions for integrability, properties of integrable functions, indefinite integral and their properties, fundamental theorem of integral calculus, mean value theorems, improper integrals, convergence at infinity, absolute and conditional convergence. Sequences and series of functions, uniform convergence of sequences and series of functions. Cantors definition of real numbers. Metric sets: Definition, real line as an example of a metric set.