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A proof question about continuity


Feb 3, 2013
Let $E⊂\mathbb{R}^{n}$ be a closed, non-empty set and $\mathbb{R}^{n}→\mathbb{R}$ be a norm. Prove that
the function
$f(x) = inf$ {$N(x-a) s.t. a∈E$}, $f :\mathbb{R}^{n}→\mathbb{R}$ is continuous and $f^{-1}(0)=E$.

(There are some hint:
$f^{-1}(0)=E$ will be implied by $E$ closed. $f :\mathbb{R}^{n}→\mathbb{R}$ is continuous implied by triangle inequality.

I still can't get the proof by the hint. So...thank you for your help!)