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Real Analysis - I (MATH209)

Review on sets and functions, review on proof techniques, finite and infinite sets, natural numbers system, countable and uncountable sets, rational numbers system, real numbers system and its properties, supremum and infimum, sequences and limits, monotone sequences, subsequences, Bolzano-Weierstrass theorem, limit supremum, limit infimum, Cauchy criterion for convergence, divergence to infinity, continuous functions, examples of discontinuity, combinations of continuous functions, continuous functions on intervals, boundednes theorem, maximum-minimum theorem, Bolzano intermediate value theorem, uniform continuity, uniform continuity theorem, monotone and inverse functions, continuous inverse theorem, derivative and its properties, example of continuous and nowhere differentiable function, chain rule, derivative of inverse function, mean value theorems and their applications, intermediate value property of derivatives., L’Hospital’s rules, Taylor’s theorem, Riemann integration, partitions and integral sums, properties of Riemann integrals, integrability of continuous and monotone functions, fundamental theorem of calculus.

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