Q:

A question about continuous functions

Let $f:B \longrightarrow \mathbb{R}^n$ a continuous and injective function from closed ball in $\mathbb{R^n}$ to $\mathbb{R}^{n}$. I'd like to know $f$ has to maps the boundary $\partial B$ in the boundary $\partial f(B)$?
Thank you.

A:

Note that $f(B)^{\circ}$, the interior of $f(B)$, is open so $f^{-1}(f(B)^{\circ})$ is an open subset of $B$. Furthermore, $f^{-1}(f(B)^{\circ}) \subseteq f^{-1}(f(B))$ and $f^{-1}(f(B)) = B$ as $f$ is injective, so $f^{-1}(f(B)^{\circ}) \subseteq B^{\circ}$. 
Now let $x \in \partial B =\overline{B}\setminus B^{\circ}$, then $x \notin f^{-1}(f(B)^{\circ})$ so $f(x) \notin f(B)^{\circ}$. However $f(x) \in f(B)$ and hence $f(x) \in \overline{f(B)}$, so $f(x) \in \overline{f(B)}\setminus f(B)^{\circ} = \partial f(B)$.

