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.


Outline syllabus

  • Introduction.
  • Limits.
  • Continuity.
  • Differentiation.
  • Integration.



Aims

  • 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.

Learning outcomes

  • 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.

Teaching methods

Lectures, problem sheets.


32 lectures, 32 tutorials

Assessment

One formal 2 hour written examination. All questions compulsory.

Full syllabus

1. Introduction
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.
2. Limits
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.
3. Continuity
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.
4. Differentiation
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.
5. Integration
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.

Reading list

Type Author(s) Title Library Blackwells Amazon
B Abbott Understanding Analysis Blackwells Amazon
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.