Why does this statement hold?

Question

I have seen the following statement:

Let $G\subset \mathbb{R}^n$ be open, bounded and $f:\overline{G}\rightarrow \mathbb{R}^n$ a continuous and open map. Then $\|f\|$ gets its maximum on the boundary of $G$.

Why does this holds? Could you explain this to me?

Answer 1

I suppose that you mean that $f|_G$ is open.

If the maximum was attained at a point $p$ ouside the boundary, then $p$ would belong to $G$. But, since $f|_G$ is open, the range of $f$ must contain an open ball centered at $p$, contradicting the assumption about the maximum.