# Oxford Analysis II (Prelims M2): Continuity and Differentiability

> A study guide for Oxford Prelims Analysis II: limits, continuity, uniform convergence, the Mean Value Theorem, L'Hôpital and Taylor, with free practice.

Source: https://practicecalculus.com/courses/university-of-oxford-analysis-ii
Updated: 2026-11-08

## At a glance

- **Course code:** M2: Analysis II (examined in Prelims paper Mathematics II; Oxford does not use numeric module codes)
- **Lectures:** 16 lectures
- **Year:** Year 1 (Prelims), after Analysis I: Sequences and Series
- **Prerequisites:** Not listed formally; it continues the study of limits and power series from Analysis I
- **Covers:** Function limits, epsilon-delta continuity, IVT, uniform continuity, uniform convergence, derivatives, Rolle, MVT, L'Hôpital, Taylor's theorem
- **Next course:** M2: Analysis III (Integration)
- **Practice here:** [Calculus 1](/learn/calculus-1) and the series units of [Calculus 2](/learn/calculus-2)

## Key takeaways

- Analysis II: Continuity and Differentiability is a 16-lecture first-year course in Oxford Mathematics Prelims, examined in paper Mathematics II.
- It proves the main theorems of single-variable differential calculus, from the Intermediate Value Theorem to Taylor's theorem.
- It also introduces uniform convergence and the Weierstrass M-test for series of functions.
- Analysis I comes before it, and Analysis III, on Riemann integration, comes after.

**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

| Syllabus topic | Practice | Read |
| --- | --- | --- |
| The function limit | [What is a limit?](/learn/calculus-1/limits/what-is-a-limit) | [Limit laws](/rules/limit-laws) |
| Squeezing limits | [Squeeze theorem](/learn/calculus-1/limits/squeeze-theorem) | [Squeeze theorem](/theorems/squeeze-theorem) |
| Continuity | [Continuity](/learn/calculus-1/limits/continuity) | [Continuity guide](/blog/continuity) |
| Intermediate Value Theorem | [Intermediate Value Theorem](/learn/calculus-1/limits/intermediate-value-theorem) | [IVT statement](/theorems/intermediate-value-theorem) |
| Maxima and minima on closed intervals | [Absolute extrema](/learn/calculus-1/applications-of-derivatives/absolute-extrema) | [Extreme Value Theorem](/theorems/extreme-value-theorem) |
| Inverse functions | [Inverse function derivatives](/learn/calculus-1/derivative-rules/inverse-function-derivatives) | [Inverse function rule](/rules/inverse-function-rule) |
| Power series and uniform convergence | [Power series and radius](/learn/calculus-2/power-series-parametric-polar/power-series-and-radius) | [Power series](/blog/power-series) |
| Definition of the derivative | [The limit definition](/learn/calculus-1/the-derivative/limit-definition) | [Limit definition of the derivative](/blog/limit-definition-of-the-derivative) |
| Differentiability | [Differentiability](/learn/calculus-1/the-derivative/differentiability) | [Product and quotient rules](/blog/product-and-quotient-rules) |
| Local extrema and Rolle's Theorem | [Critical points](/learn/calculus-1/applications-of-derivatives/critical-points) | [Fermat's theorem](/theorems/fermats-theorem), [Rolle's theorem](/theorems/rolles-theorem) |
| Mean Value Theorem | [Linear approximation and the MVT](/learn/calculus-1/applications-of-derivatives/linear-approximation-and-mvt) | [Mean Value Theorem](/theorems/mean-value-theorem) |
| L'Hôpital's formula | [L'Hôpital's rule](/learn/calculus-1/applications-of-derivatives/lhopitals-rule) | [L'Hôpital's rule](/rules/lhopitals-rule) |
| Taylor's theorem | [Taylor polynomials](/learn/calculus-2/power-series-parametric-polar/taylor-polynomials) | [Taylor's theorem](/theorems/taylors-theorem) |
| Binomial expansion | [Maclaurin series](/learn/calculus-2/power-series-parametric-polar/maclaurin-series) | [Taylor series](/blog/taylor-series) |

## 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

$$|f(x) - f(2)| = 3|x - 2| < 3\delta = \varepsilon$$

**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

$$\sin a - \sin b = \cos c\,(a - b), \qquad |\cos c| \le 1$$

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

$$e^x = 1 + x + \frac{x^2}{2} + \frac{e^{\xi}x^3}{6} \quad \text{for some } \xi \text{ between } 0 \text{ and } x$$

## 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](/blog/lhopitals-rule) and [Taylor series](/blog/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](/functions/absolute-value-function) page is a good start.
- **Review the [Mean Value Theorem guide](/blog/mean-value-theorem)** 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](/courses/university-of-oxford-introductory-calculus).

## Frequently asked questions

### What is Analysis II at Oxford?

It is the second of three Prelims analysis courses. The 2025-26 synopses say it studies continuity of functions of a real or complex variable and differentiability of functions of a real variable, proving theorems such as the Intermediate Value Theorem, Rolle's Theorem and the Mean Value Theorem.

### How is Analysis II different from school calculus?

At A level you use derivatives. In Analysis II you prove why the rules work, starting from epsilon-delta definitions of limits and continuity. The results are familiar, but the arguments are new.

### Which term is Analysis II taught in?

The course page lists 16 lectures, and the 2026 lecture notes are dated Hilary Term. The synopses themselves do not state the term, so check the current lecture timetable.

### How can I practise for Analysis II?

Get the computations completely fluent first: limits, derivative rules, L'Hôpital and Taylor polynomials. Then the proofs are about the ideas, not the algebra. The Calculus 1 path covers the computational side step by step.