MU-201 Real Analysis

Unit · year 2

MU-201 · Real Analysis

Threads space · change26 lectures6 theorems

Continuity, compactness and integration made rigorous on the real line.

PREREQUISITES

Lectures

L01	Topology of the Real Line: Open and Closed Sets —
L02	Limit Points, Closure, and Density —
L03	Compactness: Open Covers —
L04	The Heine–Borel Theorem The Heine–Borel theorem
L05	Connectedness on the Line —
L06	Limits of Functions: The ε–δ Definition —
L07	Continuity: Sequential and Topological Characterisations —
L08	The Intermediate Value Theorem The intermediate value theorem
L09	Consequences of the IVT: Roots and Fixed Points The intermediate value theorem
L10	Continuous Images of Compact Sets The Heine–Borel theorem
L11	Uniform Continuity Uniform continuity on compact sets
L12	Uniform Continuity on Compact Sets Uniform continuity on compact sets, The Heine–Borel theorem
L13	Differentiability Revisited —
L14	The Darboux Property of Derivatives —
L15	The Riemann–Darboux Integral —
L16	The Riemann Integrability Criterion The Riemann integrability criterion
L17	Classes of Integrable Functions The Riemann integrability criterion
L18	Sets of Measure Zero and Lebesgue's Criterion The Riemann integrability criterion
L19	Sequences of Functions: Pointwise versus Uniform —
L20	Uniform Limits of Continuous Functions Uniform limits of continuous functions
L21	Uniform Convergence and Integration Uniform limits of continuous functions
L22	Uniform Convergence and Differentiation Uniform limits of continuous functions
L23	Series of Functions and the Weierstrass M-Test The Weierstrass M-test
L24	Power Series on the Interval of Convergence The Weierstrass M-test
L25	Pathologies: Nowhere-Differentiable Functions The Weierstrass M-test
L26	Synthesis: Compactness, Uniformity, and Control The Heine–Borel theorem, Uniform limits of continuous functions