Autumn semester

Components of Assessment	Method(s)	Percentage weighting
Coursework	One closed book class tests  One assignment.	12.5% 12.5%
Examination	Written examination (2 hours, unseen).	75%

MS107 Real Analysis 1. (MAT1003)

The aim of this module is to extend the introduction to real analysis by studying continuity, differentiability and integration of functions of a real variable in a more formal way and hence provide a deeper understanding of those concepts. Several applications will presented alongside the theory.

At the end of the module a student should be able to:

• Prove continuity, differentiability and integrability of function by using the formal definitions and basic properties.
• Quote, prove and apply main theorems in Real Analysis (e.g., Intermediate, Extreme and Mean Value Theorems, Rolle’s Theorem, l'Hôpital's rule,Taylor’s Theorem, Fundamental Theorem of Calculus, etc.).
• Argue logically to justify proofs or give counterexamples of properties of continuity, convergence, differentiability and integrability.
• Calculate Taylor series and determine and justify its convergence and convergence of some other sequences of functions (e.g. power series).

This module contains the following topics:

• Limits of functions, continuity (ε-δ definition). Sums, products, compositions. Intermediate value theorem and extreme value theorem.
• Differentiable functions (sums, products, quotients). Differentiability implies continuity. Chain rule, inverse functions.
• Rolle's theorem, mean value theorem, l'Hôpital's rule. Higher derivatives. Taylor 's theorem.
• Theory of integration: upper and lower sums and integrals, the Riemann integral. Conditions for integrability (e.g., continuity implies integrability). Indefinite integration, and the fundamental theorem of calculus. Taylor series with integral remainder.
• Convergence of sequences of functions (e.g., Taylor series, power series).

Teaching is by lectures, tutorials and example classes. Learning takes place through lectures, exercises (example sheets), preparation for tests and background reading. The lectures follow closely the book of Howie (2001) and the students are recommended to purchase it.

3 contact hours for 10 weeks consisting of lectures, tutorials and example classes.

Recommended text:

J. M. Howie, Real Analysis, Springer (2001)
(Available in paperback from UniS bookshop, Springer or amazon.co.uk) 

J.E. Snow and K.E. Weller, Exploratory Examples for Real Analysis, Cambridge University Press (2004).
(Available in paperback from UniS bookshop, CUP or amazon.co.uk) 

Other texts:

J. Lewin, Mathematical Analysis, Cambridge University Press (2003).

P.E. Kopp, Analysis, Arnold Publishers, (1990).

S. Lang, Analysis I, Addison-Wesley (1968).

27th August 2007