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G12MAN - Mathematical Analysis

G12MAN - Mathematical Analysis (University of Nottingham). This is a series of video lectures taught by Dr. Joel Feinstein, introducing mathematical analysis building upon the experience of limits of sequences and properties of real numbers and on calculus. It includes limits and continuity of functions between Euclidean spaces, differentiation and integration. A variety of very important new concepts are introduced by investigating the properties of numerous examples, and developing the associated theory, with a strong emphasis on rigorous proof. (from unow.nottingham.ac.uk)

 Lecture 12b - Functions, Limits and Continuity

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 Workshop 01 - Mathematical Analysis Module: The Nature of the Module Revision Quiz Lecture 01 - A Revision of Notation: Sets, Subsets, Intersections and Unions Lecture 2a - Properties of the Euclidian Norm Lecture 2b - Open Balls and Closed Balls Workshop 02 - Why do We Do Proofs? Lecture 03 - Bounded Sets Lecture 4a - Examples of Bounded and Unbounded d-cells Lecture 4b - Bounded and Unbounded d-cells (cont.) Workshop 03 - Examples Class 1 Lecture 05 - Interior and Non-interior Points Lecture 06 - Interior Points/ Non-interior Points How do We Do Proofs? Part I Lecture 07 - Topology of d-dimensional Euclidian Space Lecture 8a - Open Sets and Closed Sets Lecture 8b - Sequences in d-dimensional Euclidian Space Lecture 09 - Absorption of Sequences by Sets Workshop 05 - Examples Class 2 Lecture 10a - Proof of the Sequence Criterion for Closedness Lecture 10b - Subsequences and Sequential Compactness How do We Do Proofs? Part II Lecture 11 - Subsequences of Sequences: Bolzano-Weierstrass and Heine-Borel Theorems Lecture 12a - Proof of Bolzano-Weierstrass Theorem Lecture 12b - Functions, Limits and Continuity Lecture 13a - Functions, Limits and Continuity (cont.) Lecture 13b - Continuous Functions Lecture 14a - Sequence Definition of Continuity Lecture 14b - Further Theory of Function Limits and Continuity Workshop 08 - Examples Class 4 Lecture 15 - Sandwich Theorem for Real-valued Function Limits Lecture 16 - Application of the Sandwich Theorem Lecture 17a - The Boundedness Theorem for Continuous Real-valued Functions Lecture 17b - Pointwise Convergence: Definition and Examples Lecture 18 - Sequences of Functions Lecture 19a - Uniform Convergence Lecture 19b - Rigorous Differential Calculus Lecture 20 - Fermat's Theorem, Rolle's Theorem and the Mean Value Theorem Lecture 21 - An Introduction to Riemann Integration