Construction of a continuous function which maps some point in the interior of an open set to the boundary of the Range I was studying the Inverse function theorem when I came across the following problems :

(Let the closed set $V$ i.e the range have non-empty interior)  
  
  
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*Does there exist a continuous onto function from an open set $U$ in $\mathbb{R}^n $ to a closed set $V$ in $\mathbb{R}^m$  such that some points in the interior of $U$ get mapped to the boundary of $V$?
  
*Does there exist a continuous $1-1$ map from an open set $U$ in $\mathbb{R}^n $ to a closed set $V$ in $\mathbb{R}^m$  such that some points in the interior of $U$ get mapped to the boundary of $V$?

If there are examples in $C(\mathbb{R})$ i.e continuous functions from $\mathbb{R}$ to $\mathbb{R}$, then that would be great too! Though I do need some example in the general case too.
Simpler examples will be really appreciated.
Thanks in advance.
Edit: 
The case (1) can be dealt with using any "cut-off" function.  e.g  let $U,V$ two balls around $0$ in $\mathbb{R}^n $ with radius $r(>1)$ and $1$, and be open and closed respectively.
Let $f: U \rightarrow V $ such that $x \in V \implies f(x)=x$ and $x \in U-V \implies f(x)= x/||x|| $.

The sine/cosine functions map $\mathbb{R}$ to the closed interval $[-1,1]$. 
(THIS IS WRONG: Moreover, the embedding $\mathbb{R} \to \mathbb{R}^2, x \mapsto (x, 0)$ satisfies both assertions.)
