General comments

Section H.1 General comments

Subsection H.1.1 What is examinable?

The following is copied from the Moodle page:

The set of examinable material for this unit is precisely the union of the typed lecture notes and the assigned problems on the problem sheets.

The lectures will follow the typed lecture notes. You can use them as a drop-in replacement for the lecture notes, or you can supplement one with the other. The exam will also assume that you have worked on all of the assigned problems yourself, without reference to the solutions. When I assign an exercise, I will mark it with the problem sheet number in brackets you can easily pick it out from the List of exercises. Everything else that comes up, for instance in the problems classes or in office hours, is not examinable, although it is of course my sincere hope that these additional resources will help you to learn the material! The same goes for the unassigned exercises in the lecture notes.

Subsection H.1.2 Formula sheet

The exam includes a short ‘formula sheet’ recalling some basic definitions and results from Chapter 1. This sheet is available as a pdf here

www.mileshwheeler.com/ma30252/formulasheet.pdf

. I was asked (in previous years) to limit this to a single page, and so there is not room to list all of the results even from this chapter. Do not try to read too much into which results I decided to include.

Subsection H.1.3 Using results from lectures or problem sheets

The exam begins with the following disclaimer:

“The following statement applies to all questions on this exam. In your arguments you may use any result from the lectures or the problem sheets, provided it is clear from context which result you are using. When in doubt, give a precise statement of any result that you use.”

Subsection H.1.4 About marking

Some parts of questions on the exam will ask you to recall a definition or a result. You of course do not have to use the same exact wording as in the notes, but you do need to write clearly enough that I can understand you. If you almost get the definition right, you may get partial marks.

Example H.1.

Let \((X,d)\) be a metric space, \(x_0 \in X\text{,}\) and \(f \maps X \to \R\text{.}\) State what it means for \(f\) to be continuous at \(x_0\text{.}\)

Solution 1. no marks

\begin{gather*} \abs{x-y} \lt \delta \implies \abs{f(x)-f(y)} \lt \varepsilon\text{.} \end{gather*}

Solution 2. partial marks

\begin{gather*} \forall \varepsilon \gt 0 \, \exists \delta \gt 0 \st \forall x \in X,\\ \abs{x-x_0} \lt \delta \implies \abs{f(x)-f(x_0)} \lt \varepsilon\text{.} \end{gather*}

Solution 3. full marks

\begin{gather*} \forall \varepsilon \gt 0 \, \exists \delta \gt 0 \st \forall x \in X,\\ d(x,x_0) \lt \delta \implies \abs{f(x)-f(x_0)} \lt \varepsilon\text{.} \end{gather*}

Most of the exam will instead ask you to make a mathematical argument of some kind. For these parts of the exam, just writing down a couple of relevant definitions (e.g. of terms appearing in the statement of the question) will not earn you very many marks at all. Instead, you can earn marks by showing that you have some idea about how to use these definitions and results.

Example H.2.

Let \((X,d)\) be a metric space and let \(f \maps X \to \R\) be continuous. Show that the function \(g \maps X \to \R\) defined by \(g(x)=(f(x))^2\) is also continuous.

Solution 1. no marks

A function \(f\) is continuous if

\begin{gather*} \forall \varepsilon \gt 0 \, \exists \delta \gt 0 \st \forall x \in X,\\ d_X(x,x_0) \lt \delta \implies d_Y(f(x),f(x_0)) \lt \varepsilon\text{.} \end{gather*}

Solution 2. partial marks

I’m out of time, but I would use the result from lectures about compositions of continuous functions.

Solution 3. full marks

We can write \(g = h \circ f\) where \(h \maps \R \to \R\) is the continuous map \(x \mapsto x^2\text{.}\) Thus \(g\) is a composition of two continuous functions, and hence continuous by a result from lectures.

If you have not been regularly turning in problem sheets and reading the written feedback, I strongly encourage you to look at Appendix G for some tips on writing analysis proofs.

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