# Real Analysis

## Overview

• Credit value: 15 credits at Level 6
• Convenor and tutor: Simon Hubbert
• Assessment: coursework (20%) and a two-hour examination (80%)

## Module description

In this module, which introduces the subject of mathematical analysis, we explore the theoretical underpinnings of calculus. The module is part of the Graduate Certificate in Mathematics by Distance Learning, which can serve as a qualifying course for our postgraduate degrees MSc Mathematics or MSc Mathematics and Financial Modelling. This module will cover material on sequences, series, limits, mathematically rigorous definitions of continuity, differentiation and integration, and theorems about these concepts.

The module is studied via distance learning. Each week, you will be guided through a series of learning steps. These include short instructional videos, online quizzes to test your understanding, livestreamed face-to-face examples classes, full course notes, and exercises to try at home.

### Indicative module syllabus

• Number systems: real numbers; epsilon notation
• Sequences: divergence and convergence; limits; subsequences
• Least upper bounds, greatest lower bounds, the Completeness Axiom and the Monotone Convergence Theorem
• Series: divergence and convergence; tests for divergence and convergence
• Functions: limits; continuity; tests for continuity and discontinuity
• Differentiation: Rolle’s Theorem; Mean Value Theorem; l’Hôpital’s rule
• Integration: Riemann integral; lower and upper Riemann sums
• Power series: Taylor polynomial; Taylor’s Theorem

## Learning objectives

By the end of this module, you will:

• have knowledge and understanding of, and be able to use, mathematical methods and techniques
• have knowledge and understanding of a range of results in mathematics
• appreciate the need for proof in mathematics, and be able to follow and construct mathematical arguments
• understand the importance of assumptions and have an awareness of where they are used and the possible consequences of their violation
• appreciate the power of generalisation and abstraction in the development of mathematical theories.