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For $x∈\mathbb{R}^n$ , let $B(x,r)$ denote the closed ball in $\mathbb{R}^n$(with Euclidean norm) of radius $r$ centered at $x$. Write $B=B(0,1)$.If $f,g:B→\mathbb{R}^n$ are continuous functions such that $f(x)≠g(x)$ for all $ x∈ B$, then which of the followings are true?

  1. $f(B)∩g(B)=\varnothing$
  2. There exist $ϵ>0$ such that $||f(x)-g(x)||> ϵ$ for all $ x∈ B$
  3. There exist $ϵ>0$ such that $ B(f(x), ϵ) ∩ B(g(x), ϵ)=\varnothing$ for all $ x∈ B$
  4. ${\rm int }(f(B)) ∩ {\rm int }(g(B))=\varnothing$ , where ${\rm int}(E)$ denotes the interior of a set $E$

How can I solve this problem? Can anyone help?

  • 0
    Is it supposed to read $f,g:\Bbb R^n\to\Bbb R^n$? Also, $f(x)$ and $g(x)$ are not sets, as far as I can tell here. What should $1.$ read?2012-12-20
  • 0
    sorry for my mistakes.now i have corrected it2012-12-20
  • 2
    Please consider accept the answers they give you. [How do I accept an answer?](http://meta.math.stackexchange.com/q/3286/8271)2012-12-23

2 Answers 2