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2010/1 Module Catalogue
 Module Code: MAT2004 Module Title: REAL ANALYSIS 2
Module Provider: Mathematics Short Name: MS218
Level: HE2 Module Co-ordinator: DERKS GL Dr (Maths)
Number of credits: 15 Number of ECTS credits: 7.5
Module Availability
Assessment Pattern

Unit(s) of Assessment


Weighting Towards Module Mark( %)


Coursework (1 test, 2 assignments)








Qualifying Condition(s) 


A weighted aggregate mark of 40% is required to pass this module.



Module Overview
This module builds on the level 1 module Real Analysis 1 and focuses on continuity, differentiability and integrability of real functions of one variable.
MAT1003 Real Analysis 1
Module Aims
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 be presented alongside the theory.
Learning Outcomes

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 examples or counterexamples of properties of continuity, convergence, differentiability and integrability.
Module Content
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. Contraction mapping 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.
Methods of Teaching/Learning

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.




There will be 3 contact hours for 10 weeks consisting of lectures, tutorials and example classes.
Selected Texts/Journals

Recommended texts:


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



Other texts:


  • W.F. Trench, Introduction to Real Analysis, (2003, updated February 2010). Can be downloaded free via


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


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


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


  • J.E. Snow and K.E. Weller, Exploratory Examples for Real Analysis, Cambridge University Press (2004).


Last Updated
September 10