REAL ANALYSIS 2 - 2025/6

Module code: MAT2004

Module Overview

Analysis is the branch of mathematics that rigorously studies functions, continuity, differentiability and integration. This module builds on Level 4 Real Analysis 1 (MAT1032) by studying these concepts in a more formal way and hence provides a deeper understanding of these concepts.

Module provider

Mathematics & Physics

Module Leader

GRANT James (Maths & Phys)

Number of Credits: 15

ECTS Credits: 7.5

Framework: FHEQ Level 5

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

Overall student workload

Independent Learning Hours: 59

Lecture Hours: 33

Tutorial Hours: 10

Guided Learning: 15

Captured Content: 33

Module Availability

Semester 1

Prerequisites / Co-requisites


Module content

Indicative content includes: 

  • Limits and continuity of the sums, products and composition of real-valued functions; 

  • Intermediate Value Theorem, Extreme Value Theorem; 

  • Derivatives of the sums, products, quotients, composition and inverse of real-valued functions; 

  • Rolle’s Theorem, Mean Value Theorem, L’Hôpital’s Rule; 

  • Higher derivatives, Taylor’s Theorem, Contraction Mapping Theorem; 

  • Upper and lower sums of integrals, the Riemann integral; 

  • Indefinite integration, Fundamental Theorem of Caculus, Taylor series with integral remainder. 

Assessment pattern

Assessment type Unit of assessment Weighting
School-timetabled exam/test In-semester test (50 minutes) 20
Examination Exam (2 hours) 80

Alternative Assessment


Assessment Strategy

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

  • Understanding, interpretation and manipulation of mathematical statements. 

  • Subject knowledge through implicit recall of key definitions, theorems and their proofs. 

  • Analytical ability through logical proofs or counterexamples to unseen problems in the test and exam. 

Thus, the summative assessment for this module consists of: 

  • One in-semester test taken during the semester, worth 20% of the module mark, corresponds to Learning Outcome 1, 2, 3. 

  • A synoptic examination (2 hours), worth 80% of the module mark, corresponds to Learning Outcomes 1, 2, 3. 

Formative assessment

There are two formative unassessed courseworks over an 11 week period, designed to consolidate student learning. 


Individual written feedback is provided to students for formative unassessed courseworks. The feedback is timed such that feedback from the first coursework will assist students with preparation for the in-semester test. The feedback from both courseworks and the in-semester test will assist students with preparation for the synoptic examination. Students also receive verbal feedback during lectures and tutorials. 

Module aims

  • Extend students' understanding of real-valued functions by studying their properties in a formal way.
  • Enable students to determine the properties of a given function from first principles.
  • Facilitate students' understanding of analysis through frequently encountered examples, counterexamples, problems and applications.

Learning outcomes

Attributes Developed
001 Students will understand real-valued functions and be able to prove limits, continuity, differentiability and integrability using the formal definitions and basic properties. KC
002 Students will be able to quote, prove and apply main theorems, including (but not limited to) the Intermediate Value Theorem, Extreme Value Theorem, Rolle's Theorem, Mean Value Theorem, L'Hôpital's Rule, Taylor's Theorem and Fundamental Theorem of Calculus. KC
003 Students will be able to give logical proofs or counterexamples to unseen statements relating to limits, continuity, differentiability and integrability. KCT

Attributes Developed

C - Cognitive/analytical

K - Subject knowledge

T - Transferable skills

P - Professional/Practical skills

Methods of Teaching / Learning

The learning and teaching strategy is designed to:

Give a detailed introduction to real-valued functions, limits, continuity, derivatives and integrals and ensure experience in the methods used to interpret, understand and solve problems in analysis.

The learning and teaching methods include:

  • Three one-hour lectures per week for eleven weeks, with typeset notes to complement the lectures. The lectures provide a structured learning environment with opportunities for students to ask questions and to practice methods taught. 

  • Ten tutorials for guided discussion of solutions to problem sheets (provided to students in advance for completion to reinforce their understanding and guide their learning). 

  • Two unassessed courseworks to provide students with further opportunity to consolidate learning. Students receive individual written feedback on these as guidance on their progress and understanding.   

  • Lectures may be recorded or equivalent recordings of lecture material provided. These recordings are intended to give students the opportunity to review parts of lectures that they may not fully have understood and should not be seen as an alternative to attending lectures. 

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
Upon accessing the reading list, please search for the module using the module code: MAT2004

Other information

The School of Mathematics and Physics is committed to developing graduates with strengths in Digital Capabilities, Employability, Global and Cultural Capabilities, Resourcefulness and Resilience, and Sustainability. This module is designed to allow students to develop knowledge, skills, and capabilities in the following areas:  

  • Digital Capabilities: The SurreyLearn page for MAT2004 features a dynamic discussion forum where students can pose questions and engage with others using e.g. LaTeX and MathML tools. This enhances their digital competencies and facilitates collaborative learning and information sharing.

  • Employability: The module MAT2004 equips students with skills which significantly enhance their employability. Students gain mathematical proficiency, which hones critical thinking and problem-solving abilities. Students learn to evaluate complex problems, break them into manageable components, and apply logical reasoning to arrive at solutions — these are highly sought after skills in any profession.

  • Global and Cultural Capabilities: Students enrolled in MAT2004 originate from a variety of countries and have a wide range of cultural backgrounds. Students are encouraged to work together during problem-solving teaching activities in tutorials and lectures, which naturally facilitates the sharing of different cultures.

  • Resourcefulness and Resilience: MAT2004 is a module which demands a rigorous approach to Real Analysis, to which students will learn to adapt. They will gain skills in analysing problems and lateral thinking. Students will complete assessments which challenge them and build resilience. 

Programmes this module appears in

Programme Semester Classification Qualifying conditions
Mathematics with Statistics BSc (Hons) 1 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics BSc (Hons) 1 Compulsory A weighted aggregate mark of 40% is required to pass the module
Mathematics MMath 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 2025/6 academic year.