Limits and Continuity

Let the function $f : \mathbb {R}^{n} \rightarrow \mathbb {R}^{n}$ satisfy the following two conditions:

1. $f(K)$ is compact whenever $K$ is a compact subset of $\mathbb {R}^{n}$.

2. If $\left\{ K_n\right\}$ is a decreasing sequence of compact subsets of $\mathbb {R}^{n}$, then

   $$f\left(\bigcap _1^{\infty } K_n\right) = \bigcap _1^{\infty } f(K_n).$$

Prove that $f$ is continuous.

Prove that a map $g : \mathbb {R}^{n} \rightarrow \mathbb {R}^{n}$ is continuous only if its graph is closed in $\mathbb {R}^{n} \times \mathbb {R}^{n}$. Is the converse true?

See also Problem 1.2.14.

Let $U \subset \mathbb {R}^{n}$ be an open set. Suppose that the map $h : U \rightarrow \mathbb {R}^{n}$ is a homeomorphism from $U$ onto $\mathbb {R}^{n}$, which is uniformly continuous. Prove $U = \mathbb {R}^{n}$.

Let $f$ be a real valued function on $\mathbb {R}^{2}$ with the following properties:

Let $f$ be a continuous function from the ball $B_n = \left\{ x \in \mathbb {R}^{n} \mid \left\| x\right\| < 1\right\}$ into itself. (Here, $\left\| \cdot \right\|$ denotes the Euclidean norm.) Assume $\left\| f(x)\right\| < \left\| x\right\|$ for all nonzero $x \in B_n$. Let $x_0$ be a nonzero point of $B_n$, and define the sequence $(x_k)$ by setting $x_k = f(x_{k-1})$. Prove that $\lim x_k = 0$.

Let $N$ be a norm on the vector space $\mathbb {R}^{n}$; that is, $N : \mathbb {R}^{n} \rightarrow \mathbb {R}$ satisfies

for all $x, y \in \mathbb {R}^{n}$ and $\lambda \in \mathbb {R}$.

A map $f : \mathbb {R}^{m} \rightarrow \mathbb {R}^{n}$ is proper if it is continuous and $f^{-1}(B)$ is compact for each compact subset $B$ of $\mathbb {R}^{n}$; $f$ is closed if it is continuous and $f(A)$ is closed for each closed subset $A$ of $\mathbb {R}^{m}$.

Suppose that $F : \mathbb {R}^{n} \rightarrow \mathbb {R}^{n}$ is continuous and satisfies

for all $x, y \in \mathbb {R}^{n}$ and some $\lambda > 0$. Prove that $F$ is one-to-one, onto, and has a continuous inverse.

See also Problem 1.2.12.