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MAT2004 Real Analysis II

New webpages

More details can be found via SurreyLearn. If you are a student on this module, you will find it in your list of courses.


In Autumn 2013, the Real Analysis 2 module will be taught by

Dr James Grant .

General description of the module

This module considers functions on the real line. First we will look at the formal definition of a limit of a function, strengthening our intuition and building on the definition of a sequence of real numbers. With this foundation, we explore concepts as continuity, differentiability and integrability. Again, intuitively we know these concepts, but in this course we will formalise this intuition and see that sometimes this can give rise to surprising observations.

Comparision with other modules

The module has a similar flavour as the latter part of Real Analysis 1. It provides the theoretical justification of various methods you are using in Calculus, ODEs, etc. Proofs are a key part of this module, but there are also many examples and some methods that follow on from the general theory.

Preliminary reading material

  • MAT1032 Real Analysis 1: notes on bounds (§9), limits of sequences (§11-13);
  • MAT1030 Calculus: material on functions, differentiation, and integration;
  • J.M. Howie, “Real Analysis”, Springer, 2001,  (available via the university subscription with Springer): §2.1-2; §3.1-2.
  • K. Hirst, “Calculus on one variable”, Springer, 2006,  (avialable via the university subscription with Springer). This book has most material from Calculus.


Class Details

Summaries and extra material

Summaries of the main topics will be added to this website during the semester.



The module is assessed by one class test, roughly midway through the semester (25%) and a final examination (75%).

The test is preceeded an unassessed assignment and there will also be an unassesse assignment near the end of the semester to help you prepare for the examination.


Every even week,  there will be a short quiz. The quizes and answers are posted here afterwards.


Each chapter of the notes has an associated exercise sheet. The exercise sheets will help your learning and revision for this module. You are strongly advised to attempt the questions in the exercise sheets, see suggesions below. Some of the questions in the assignments/test/examination will be very similar to the ones in the exercise sheets.

At this moment the following exercise sheets are available:

Selected Texts

  • Other textbooks which can be used for background reading:
    • J.E. Snow and K.E. Weller, Exploratory Examples for Real Analysis, Cambridge University Press (2004).
    • J. Lewin, Mathematical Analysis, Cambridge University Press (2003).
    • P.E. Kopp, Analysis, Arnold Publishers, (1990).
    • S. Lang, Analysis I, Addison-Wesley (1968).

Past Examination Papers

Past examination papers for this module can be found here or by following the link on the left hand menu. Hard copies of last year's exam paper are available from the lecturer and will be handed out during the later lectures.