MAT209 REAL ANALYSIS

3 Hours/Week, 3 Credits

Sets: Functions; concept of measurability, simple function, elementary properties of measures, outer measures, measurable sets and Lebesgue measure, non-Lebesgue measurable sets; relations; equivalence relations; countable, compact and connected sets. Real number system: Supremum and infimum; the completeness axiom and Dedekindís axiom; open and closed sets; limit/cluster points. Sequence: Definition and convergence of a sequence; monotonic sequence; bounded sequence; Cauchy sequence. Infinite series: Concept of sum and convergence; series of positive terms; alternating series; absolute and conditional convergence; tests for convergence. Limit and continuity: Limit and continuity of functions with their properties; uniform continuity; Heine-Borel theorem. Derivatives: Differentiability of functions; Rolleís theorem; mean value theorem; Taylorís theorem with remainder in Lagrangeís and Cauchyís forms; Maclaurinís series; expansion of functions. Power series: Interval and radius of convergence; differentiation and integration of power series; identity theorem; Abel's continuity theorem. Riemann integration: Riemann sum and Riemann integral; Darboux sums and Darboux integrals, Darboux integrability and Riemann integrability; fundamental theorem of integral calculus. Improper integrals and tests for their convergence.