School of Mathematics and Statistics (SoMaS)

MAS221 Analysis

This course is a foundation for the rigorous study of continuity and convergence of functions, both in one and in several variables. It begins with the theoretical underpinnings of calculus, and goes on to apply them to sets in which the `points' are themselves functions. Applications are included, and examples which demonstrate the need for rigour and logical precision.

Prerequisites: MAS111 (Mathematics Core II); MAS114 (Numbers and Groups)
Corequisites: MAS211 (Advanced Calculus and Linear Algebra)

The following modules have this module as a prerequisite:

Outline syllabus

• Supremum/infimum, Completeness Axiom
• Limits of sequences, Cauchy sequences
• Limits of functions
• Continuity of functions
• Differentiation
• Series
• Integration
• Limits of sequences of functions
• The uniform convergence theorem and applications

Aims

1. To introduce students properly to rigorous analysis.
2. To help students appreciate the necessity of rigour through the use of examples.
3. To show students the mathematical beauty of the principles in the development of calculus and its generalisations.
4. To give students sufficient foundations in analysis to enable them to study further analysis modules, and modules which make use of analysis.
5. To show students significant applications of the theory developed in the module.

Learning outcomes

1. Appreciate the theoretical underpinnings of calculus in one variable, and be able to rigorously derive and use the main results.
2. Understand and be able to use notions of convergence involving sequences of functions, including the difference between pointwise and uniform convergence.
3. Apply the Weierstrass M-test and the uniform convergence theorem for integrals to examples.

Teaching methods

Lectures, tutorials, problem solving

Assessment

90% exam: This is taking the form of a Blackboard test this year. It will be designed to take one hour and be open for a 24 hour period. The examinable material is what is contained in the Semester 1 notes and the updated Semester 2 notes.

Full syllabus

There may be some variation in the detail of the syllabus and the number of lectures for each section.

Semester 1

• Inequalities, supremum and infimum, completeness axiom of the real numbers, examples. (5 lectures)
• Sequences, formulation of ε− N definition of limit, discussion and examples. Algebra of limits. Convergence of bounded monotone sequences. Subsequences. Bounded sequences have convergent subsequences. Sandwich rule. Cauchy sequences, proof that a sequence converges if and only if it is Cauchy. (9 lectures)
• The ε− δ formulation of limit of a real function, and continuity. Discussion and examples. Intermediate value theorem. Continuous functions on closed bounded intervals attain their maxima. Limits of functions and continuity in terms of sequences. (8 lectures)

Semester 2

• Definition of differentiation in terms of a limit. Proof that any differentiable function is continuous. Examples and counterexamples. Proof of sum, product and chain rules. Rolle's theorem. Mean value theorem. Further examples and applications. (6 lectures)
• Convergence of series in terms of the sequence of partial sums. Examples, including geometric series and harmonic series. Tests for convergence: comparison test, absolute convergence, ratio test. (4 lectures)
• Definition of the Riemann integral. Basic properties. Examples. Continuous functions are Riemann integrable. The fundamental theorem of calculus. Improper integrals. (4 lectures)
• Sequences and series of functions; pointwise and uniform convergence. Proof that a uniform limit of a sequence of continuous functions is continuous. Example to show the same is not true of pointwise limits. Uniform continuity. The uniform convergence theorem for integrals, and application to differentiation. Series of functions and the Weierstrass M-test for uniform convergence. Example of an everywhere continuous but nowhere differentiable function. (6 lectures)
• Applications; power series, radius of convergence, termwise differentiation. Exponential function and e. (2 lectures)

Reading list