Quiz XI.

Let $f$ and $g$ be real-valued functions on a subset $S$ of $\mathbb{R}^n$ and $g$ continuous on $S$.

(a) Let $n=1$ and $S$ be compact. If $g(S)=S$, then $g(c)=c$ for some $c \in S$.

(b) Let $n=1$ and $S=\mathbb{R}$. If $G(f)$ is closed, then $f$ is continous on $S$.

(c) Let $n=1$ and $S$ be connected. If $f$ is one-to-one on $S$ and $f^{-1}$ is a continuous real-valued function on $f(S)$, then $f$ is continuous on $S$.

(d) Let $\inf g(S) \le 4.8 \le g(c)$ for some $c \in S$. If $S$ is connected, then $4.8 \in g(S)$.

(e) Let $G(f)$ be closed. Then $f$ is continuous at a point $c \in S$ if and only if there is a neighborhood $U$ of $c$ on which $f$ is bounded.

(f) Let $|g|\le 4.8$ on $S$. If $S$ is compact, then $\{x\in S| |g(x)| = 4.8\}$ is compact.

(g) Let $n=1$ and $g(S) \subset S$. Then $g$ need not be bounded on a nonempty bounded subset of $S$.

(h) Let $n > 1$ and $A=\{x=(x_1, \cdots, x_n) \in \mathbb{R}^n | \|x\| + \sum _{i=1} ^{n-1} |x_i| = 4.8 \}$. If $A\subset S$, then $g(A)$ is a compact interval.

(i) Let $n=1$ and $S=[-p, p]$ where $p \in \mathbb{N}$. If $g(S) = S$ and $g(-p) = g(0) = g(p)$, then there exist $a,b,c \in S$ with $a<b<c$ such that $g(a) = g(b) = g(c) = 0$.

(j) Let $n=1$ and $|x|+|g(x)| \le 4.8$ for all $x \in S$. If every decreasing sequence in $S$ has a cluster point in $S$, then $g(S)$ is compact.

(k) Let $n=1$ and $g(S) \subset S$. If $S$ is connected and $S \cap bd(S) \ne \emptyset$, then there is $c \in S$ such that $g(c) = c$.

(l) If $A$ is a nonempty subset of $S$ and $cl(A)$ is compact, then for any $\epsilon>0$, there exist $x_1, \cdots, x_p \in A$ such that $g(A) \subset \bigcup _{i=1} ^p N(g(x_i);\epsilon)$.

(m) If $g(S) \subset \mathbb{R}\backslash\mathbb{Q}$ and $G(g)$ compact and infinite, then $g$ is not one-to one on $S$.