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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

Go to the Course Home or watch other lectures:

Workshop 01 - Mathematical Analysis Module: The Nature of the Module

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