| course |
course_year |
question_number |
tags |
title |
year |
Analysis II |
IB |
2 |
|
Paper 4, Section I, F |
2018 |
State the Bolzano-Weierstrass theorem in $\mathbb{R}$. Use it to deduce the BolzanoWeierstrass theorem in $\mathbb{R}^{n}$.
Let $D$ be a closed, bounded subset of $\mathbb{R}^{n}$, and let $f: D \rightarrow \mathbb{R}$ be a function. Let $\mathcal{S}$ be the set of points in $D$ where $f$ is discontinuous. For $\rho>0$ and $z \in \mathbb{R}^{n}$, let $B_{\rho}(z)$ denote the ball $\left{x \in \mathbb{R}^{n}:|x-z|<\rho\right}$. Prove that for every $\epsilon>0$, there exists $\delta>0$ such that $|f(x)-f(y)|<\epsilon$ whenever $x \in D, y \in D \backslash \cup_{z \in \mathcal{S}} B_{\epsilon}(z)$ and $|x-y|<\delta$.
(If you use the fact that a continuous function on a compact metric space is uniformly continuous, you must prove it.)