Oxford Analysis II (Prelims M2): Continuity and Differentiability

Analysis II: Continuity and Differentiability is the second analysis course in the Oxford Mathematics Prelims, the first year of the BA/MMath. It has 16 lectures and is examined in paper Mathematics II. Where school calculus teaches you to use derivatives, Analysis II proves why they behave the way they do. This guide summarises the official 2025-26 Prelims synopses and links each topic to free practice, so you can make the calculations automatic and save your energy for the proofs.

What Oxford Analysis II covers

The synopses say the course studies continuity of functions of a real or complex variable and differentiability of functions of a real variable. By the end you should be able to prove results such as the Intermediate Value Theorem, Rolle's Theorem and the Mean Value Theorem, and you continue the study of power series from Analysis I.

The syllabus runs in three blocks:

1. Limits and continuity. The function limit, continuity on subsets of \mathbb{R} and \mathbb{C} in terms of \varepsilon and \delta, the algebra of continuous functions, the Intermediate Value Theorem, boundedness and attainment of maxima and minima on closed intervals, uniform continuity, and the Continuous Inverse Function Theorem for monotone functions.
2. Sequences and series of functions. Uniform convergence, the Weierstrass M-test, continuity of a uniform limit, and continuity of functions defined by power series.
3. Differentiability. The derivative, the algebra of derivatives, derivatives of inverse functions and of power series, Rolle's Theorem, the Mean Value Theorem and Cauchy's Mean Value Theorem with applications, L'Hôpital's formula, Taylor's theorem with Lagrange remainder, and the binomial expansion with arbitrary index.

Compared with a North American sequence, the results are those of Calculus 1 plus Taylor series from Calculus 2, but every one of them is proved rather than used.

Analysis II topic map

Example problems at the Analysis II level

1. An epsilon-delta proof. Show f(x) = 3x + 1 is continuous at x = 2. Given \varepsilon > 0, take \delta = \varepsilon/3. Then |x - 2| < \delta gives

2. Using the Mean Value Theorem. Show |\sin a - \sin b| \le |a - b|. By the MVT there is a c between a and b with

3. Taylor with remainder. For e^x about 0, Lagrange's form gives

How to do well in Analysis II

• Learn the definitions word for word. Most proofs start by writing out what continuity or differentiability means and then aiming for it.
• Make the calculations automatic. If L'Hôpital's rule and Taylor series are second nature, you can focus on why each step is justified.
• Collect counterexamples. Functions like |x| (continuous, not differentiable) are worth knowing well. The absolute value function page is a good start.
• Review the Mean Value Theorem guide before exams. It drives a large share of the applications in the course.

Tip: Analysis II feeds straight into Analysis III, where integration is made rigorous and the Fundamental Theorem of Calculus is proved. For the applied side of first-year calculus, see the Oxford Introductory Calculus guide.