Section: Course Materials | M2: Analysis II - Continuity and Differentiability (2026-27)

Lecturer: Profile: Richard Earl

Course information

Course term: Hilary

Course lecture information: 16 lectures

Course overview:

In this term's lectures, we study continuity of functions of a real or complex variable, and differentiability of functions of a real variable.

Learning outcomes:

At the end of the course students will be able to apply limiting properties to describe and prove continuity and differentiability conditions for real and complex functions. They will be able to prove important theorems, such as the Intermediate Value Theorem, Rolle's Theorem and Mean Value Theorem, and will continue the study of power series and their convergence.

Course synopsis:

Definition of the function limit. Definition of continuity of functions on subsets of \(\mathbb{R}\) and \(\mathbb{C}\) in terms of \(\varepsilon\) and \(\delta\). The algebra of continuous functions; examples, including polynomials. Intermediate Value Theorem for continuous functions on intervals. Boundedness, maxima, minima and uniform continuity for continuous functions on closed intervals. Monotone functions on intervals and the Continuous Inverse Function Theorem.

Sequences and series of functions, uniform convergence. Weierstrass's M-test for uniformly convergent series of functions. Uniform limit of a sequence of continuous functions is continuous. Continuity of functions defined by power series.

Definition of the derivative of a function of a real variable. Algebra of derivatives, examples to include polynomials and inverse functions. The derivative of a function defined by a power series is given by the derived series (proof not examinable). Vanishing of the derivative at a local maximum or minimum. Rolle's Theorem, Mean Value Theorem, and Cauchy's (Generalized) Mean Value Theorem with applications: Constancy Theorem, monotone functions, exponential function and trigonometric functions. L'Hôpital's Formula. Taylor's Theorem with remainder in Lagrange's form; examples. The binomial expansion with arbitrary index.

Section outline

- Select activity Introduction and Chapter 1
  
  Introduction and Chapter 1 File
  
  We discuss limits and continuity of functions - initially real functions on real intervals, and then generalizing to other domains.
- Select activity Chapter 2
  
  Chapter 2 File
  
  We meet several important theorems from the 19th century about continuous real functions on intervals.
- Select activity Chapter 3
  
  Chapter 3 File
  
  We discuss pointwise and uniform convergence of functions, showing further that power series define continuous functions on their disc of convergence.
- Select activity Chapter 4
  
  Chapter 4 File
  
  We define differentiability, prove Leibniz's rules on the algebra of differentiation, and some early results - Fermat, Rolle, MVT, DIFT.
- Select activity Sheet 1
  
  Sheet 1 Assignment
  
  Continuity and Limits on Intervals. This covers material from the first two lectures.
- Select activity Sheet 2
  
  Sheet 2 Assignment
  
  We extend the notion of continuity to general domains in R and C. Then introduce limit points so that limits can be defined on general domains too.
- Select activity Sheet 3
  
  Sheet 3 Assignment
  
  We meet some first theorems about continuous functions on a closed bounded interval: the IVT and the boundedness theorem.
- Select activity Sheet 4
  
  Sheet 4 Assignment
  
  Problems concerning the invertibility of continuous functions and uniform continuity.
- Select activity Sheet 5
  
  Sheet 5 Assignment
  
  Problems largely on pointwise and uniform convergence, with some introductory exercises on differentiability.

Main content blocks

Section outline