REAL ANALYSIS 2 - 2018/9

Module code: MAT2004

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.

Module provider


Module Leader

GRANT JD Dr (Maths)

Number of Credits: 15

ECTS Credits: 7.5

Framework: FHEQ Level 5

JACs code: G100

Module cap (Maximum number of students): N/A

Module Availability

Semester 1

Prerequisites / Co-requisites

MAT1032 Real Analysis 1

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.

Assessment pattern

Assessment type Unit of assessment Weighting
Examination EXAMINATION - 2 HOURS 80
School-timetabled exam/test IN-SEMESTER TEST (50 MINS) 20

Alternative Assessment


Assessment Strategy

The assessment strategy is designed to provide students with the opportunity to demonstrate:

·         Understanding of and ability to interpret and manipulate mathematical statements. 

·         Subject knowledge through the recall of key definitions, theorems and their proofs.

·         Analytical ability through the solution of unseen problems in the test and exam.


Thus, the summative assessment for this module consists of:

·         One two hour examination (three best questions out of four contribute to exam mark) at the end of Semester 1; worth 80% module mark.

·         One In-semester test; worth 20% module mark.


Formative assessment and feedback


Students receive written feedback via marked coursework assignments over an 11 week period.  In addition, verbal feedback is provided by lecturer/class tutor at weekly tutorial lectures. 

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

Attributes Developed
1 Prove continuity, differentiability and integrability of function by using the formal definitions and basic properties. KC
2 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.).  KC
3 Argue logically to justify proofs or give examples or counterexamples of properties of continuity, convergence, differentiability and integrability. KCT

Attributes Developed

C - Cognitive/analytical

K - Subject knowledge

T - Transferable skills

P - Professional/Practical skills

Overall student workload

Lecture Hours: 36

Tutorial Hours: 12

Methods of Teaching / Learning

The learning and teaching strategy is designed to provide:

A detailed introduction to continuity, differentiability and integrability of real-valued functions.
Experience (through demonstration) of the methods used to interpret, understand and solve problems in analysis

The learning and teaching methods include:

3 x 1 hour lectures per week x 11 weeks, with projector-displayed written notes to supplement the module handbook and Q + A opportunities for students.

1 x 1 hour interactive problem solving session/tutorial lecture per week x 11 weeks.


Indicated Lecture Hours (which may also include seminars, tutorials, workshops and other contact time) are approximate and may include in-class tests where one or more of these are an assessment on the module. In-class tests are scheduled/organised separately to taught content and will be published on to student personal timetables, where they apply to taken modules, as soon as they are finalised by central administration. This will usually be after the initial publication of the teaching timetable for the relevant semester.

Reading list

Reading list for REAL ANALYSIS 2 :

Programmes this module appears in

Programme Semester Classification Qualifying conditions
Mathematics BSc (Hons) 1 Compulsory A weighted aggregate mark of 40% is required to pass the module

Please note that the information detailed within this record is accurate at the time of publishing and may be subject to change. This record contains information for the most up to date version of the programme / module for the 2018/9 academic year.