| course |
course_year |
question_number |
tags |
title |
year |
Analysis II |
IB |
3 |
|
Paper 1, Section II, F |
2018 |
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 $a \in U$.