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This paper is an introduction to the basic techniques of real analysis in the familiar context of single-variable calculus.

Analysis is, broadly, the part of mathematics which deals with limiting processes. The main examples students have met in school and first-year university are from calculus, where the derivative and integral are defined using quite different limiting processes. Real analysis specialises to real-valued functions of a real variables. The methods of analysis have been developed over the past two centuries to give mathematicians rigorous methods for deciding whether a formal calculation is correct or not. This paper discusses the basic ideas of analysis. At the end of the semester, students should have a grounding in the methods of analysis which will prove invaluable in later years.

Paper title Real Analysis
Paper code MATH201
Subject Mathematics
EFTS 0.1500
Points 18 points
Teaching period First Semester
Domestic Tuition Fees (NZD) $904.05
International Tuition Fees (NZD) $3,954.75

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MATH 170
MATH 353
Schedule C
Arts and Music, Science
MATH 201 is compulsory for the Mathematics major and is of particular relevance, also, for students majoring in Statistics, Physics or any discipline requiring a quantitative analysis of systems and how they change with space and time.
Teaching staff
To be advised
Paper Structure

This paper will begin by developing an axiomatic description of the real line. We will then use this one-dimensional construction to develop the n dimensional Euclidean spaces and understand their properties. The bulk of this course will be taken up with understanding rigorous definitions of limits in a variety of settings.

Main topics:

  • A review of the real number system
  • The completeness axiom
  • The Euclidean spaces
  • The distance function and open and closed sets
  • Limits of sequences and the algebra of limits
  • Limits of series and the algebra of limits
  • Continuous functions
  • Limits of functions and the algebra of limits
  • Applications of real analysis in one dimensional calculus
Textbooks are not required for this paper.
Course outline
View course outline for MATH 201
Graduate Attributes Emphasised
Critical thinking.
View more information about Otago's graduate attributes.
Learning Outcomes

On completion of your study of MATH 201 you are expected to:

  • Understand the formal definition of Euclidean spaces. Particularly the real number line and plane.
  • Understand the notion of open and closed sets in the Euclidean setting.
  • Understand the rigorous definition of convergence for a sequence or series and apply appropriate tools determine whether example sequences/series are convergent.
  • Understand the definition of a continuous function and the key properties of such functions.
  • Understand convergence of functions both pointwise and uniform.

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First Semester

Teaching method
This paper is taught On Campus
Learning management system


Stream Days Times Weeks
L1 Monday 12:00-12:50 9-13, 19-22
Wednesday 12:00-12:50 9-12, 18-22
Friday 09:00-09:50 9-12, 18-22


Stream Days Times Weeks
Attend one stream from
T1 Thursday 11:00-11:50 10-12, 18-22
T2 Thursday 14:00-14:50 10-12, 18-22