| course |
course_year |
question_number |
tags |
title |
year |
Analysis II |
IB |
6 |
|
Paper 2, Section II, E |
2012 |
Let $f: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ be a mapping. Fix $a \in \mathbb{R}^{n}$ and prove that the following two statements are equivalent:
(i) Given $\varepsilon>0$ there is $\delta>0$ such that $|f(x)-f(a)|<\varepsilon$ whenever $|x-a|<\delta$ (we use the standard norm in Euclidean space).
(ii) $f\left(x_{n}\right) \rightarrow f(a)$ for any sequence $x_{n} \rightarrow a$.
We say that $f$ is continuous if (i) (or equivalently (ii)) holds for every $a \in \mathbb{R}^{n}$.
Let $E$ and $F$ be subsets of $\mathbb{R}^{n}$ and $\mathbb{R}^{m}$ respectively. For $f: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ as above, determine which of the following statements are always true and which may be false, giving a proof or a counterexample as appropriate.
(a) If $f^{-1}(F)$ is closed whenever $F$ is closed, then $f$ is continuous.
(b) If $f$ is continuous, then $f^{-1}(F)$ is closed whenever $F$ is closed.
(c) If $f$ is continuous, then $f(E)$ is open whenever $E$ is open.
(d) If $f$ is continuous, then $f(E)$ is bounded whenever $E$ is bounded.
(e) If $f$ is continuous and $f^{-1}(F)$ is bounded whenever $F$ is bounded, then $f(E)$ is closed whenever $E$ is closed.