MAS292 Continuity and Integration (NJTech)
|Semester 1, 2016/17||10 Credits|
|Lecturer:||Dr Simon Willerton||Reading List|
|Aims||Outcomes||Teaching Methods||Assessment||Full Syllabus|
The calculus of A-level and first year courses is sufficient for many intermediate applications, but quite inadequate as a foundation for more advanced studies. The theory of Fourier series shows its shortcomings very well, and generalisations of the calculus to infinite-dimensional spaces (which are very valuable in clarifying more difficult problems) are impossible without a more accurate appreciation of how single variable calculus works. This course provides just such a rigorous analysis of single variable calculus and, in doing so, challenges the imagination with weird examples showing how strange functions can be, while still being amenable to study.
There are no prerequisites for this module.
No other modules have this module as a prerequisite.
- To give students an understanding of and facility with rigorous real analysis, and an appreciation of the need for rigour.
- To give students an understanding of integration defined in terms of areas and its relation to integration as the inverse of differentiation.
- be able to follow rigorous proofs in real analysis;
- be able to construct simple ϵ−N proofs;
- be able to prove continuity/discontinuity of given functions;
- be able to prove differentiability or otherwise of given functions;
- be able to construct simple examples of badly behaved functions;
- to understand the definitions of derivative and integral and the relationship between these ideas.
Lectures, problem sheets.
32 lectures, 32 tutorials
One formal 2 hour written examination. All questions compulsory.
Limiting processes in mathematics, e.g., differentiation and infinite sums. Motivation (as much as possible) for a more systematic treatment. Real numbers as limits of truncated decimal expansions. Supremum/infimum, completeness axiom.
Sequences and subsequences. Limits of sequences: formulation in terms of ϵ−N with discussion. Algebra of limits. Convergence of bounded monotone sequences. Bounded sequences have convergent subsequences. Sandwich rule.
Functions. Continuity, in terms of sequences. Intermediate Value Theorem. Continuous functions on closed bounded intervals attain their maxima. The limit of a function and mention of ϵ−δ definition of continuity.
Definition in terms of a limit. Examples of non-differentiable functions, and a differentiable but not twice-differentiable function. Proof of sum, product and chain rules. Rolle's Theorem. Mean Value Theorem.
Definition of the Riemann integral. Example of a non-integrable function. Proof that continuous functions are Riemann integrable. Relation of integration to differentiation. Examples of discontinuous, integrable functions.
|B||Binmore||Mathematical Analysis: a Straightforward Approach||Blackwells||Amazon|
|B||Bryant||Yet another Introduction to Analysis||Blackwells||Amazon|
|C||Rudin||Principles of mathematical analysis||Blackwells||Amazon|
(A = essential, B = recommended, C = background.)
Most books on reading lists should also be available from the Blackwells shop on Mappin Street.