Paper 1, Section II, F

Let $U \subset \mathbb{R}^{n}$ be a non-empty open set and let $f: U \rightarrow \mathbb{R}^{n}$.

(a) What does it mean to say that $f$ is differentiable? What does it mean to say that $f$ is a $C^{1}$ function?

If $f$ is differentiable, show that $f$ is continuous.

State the inverse function theorem.

(b) Suppose that $U$ is convex, $f$ is $C^{1}$ and that its derivative $D f(a)$ at a satisfies $\|D f(a)-I\|<1$ for all $a \in U$, where $I: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ is the identity map and $\|\cdot\|$ denotes the operator norm. Show that $f$ is injective.

Explain why $f(U)$ is an open subset of $\mathbb{R}^{n}$.

Must it be true that $f(U)=\mathbb{R}^{n}$ ? What if $U=\mathbb{R}^{n}$ ? Give proofs or counter-examples as appropriate.

(c) Find the largest set $U \subset \mathbb{R}^{2}$ such that the map $f: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ given by $f(x, y)=\left(x^{2}-y^{2}, 2 x y\right)$ satisfies $\|D f(a)-I\|<1$ for every $a \in U$.

*Typos? Please submit corrections to this page on GitHub.*