Let $U \subseteq \mathbb{R}^n$ open subset of $\mathbb{R}^n$ and $f:U\to\mathbb{R}^n$ be a $C^\infty$– function. Suppose for every $x\in U$ the derivative of at $x$, $df_x$ is non singular. Which of the following are true ?
$1)$ If $V \subset U$ is open then $f(V)$ is open in $\mathbb{R}^n$
$2)$ $f:U\to f(U)$ is a homeomorphism
$3)$ $f$ is one-one
$4)$ If $V \subset U$ is closed then $f(V)$ is closed in $\mathbb{R}^n$
As a consequence of inverse function theorem, $1)$ is true infact $f(U)$ is open in $\mathbb{R}^n$ and $F(x,y)=\begin{bmatrix}e^x\cos y \\ e^x \sin y\end{bmatrix}$
for $F:\mathbb{R}^2\to\mathbb{R}^2$ is a counter example to $2)$ and $3)$ since $F$ is not injective infact periodic hence not a homeomorphism.
For $4)$ if we consider $V=U$ then it is false. But I guess we should not assume $V=U$ since then $V\subsetneq U$. I am not sure if $\subsetneq$ and $\subset$ mean the same here.
I am unable to show $4)$ is not true for a proper closed subset of $U$. Any help will be appreciated.
Best Answer
Answer for 4): Let $V=\mathbb N$ in $\mathbb R$ and $f(x)=\arctan x$. Then $\frac {\pi} 2$ is a limit point of $f(V)$ which does not belong to $f(V)$.