Is  the following statement is True/false ?  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   There exist $ϵ&gt;0$ such that $ B(f(x), ϵ) ∩ B(g(x), ϵ)=\varnothing$  for all $ x∈ B$  My attempt :  i take  $f(x) = x$, $g(x) = x+1$, now  i visualize the diagram  i take $ϵ$ as constant  that greater then $0$  from  the diagram  i can conclude that this statement is false .  Is  its  True ?  any hints/solution will be appreciated  thanks u
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?       $f(B)∩g(B)=\varnothing$  There exist $ϵ&gt;0$ such that $||f(x)-g(x)||&gt; ϵ$ for all $ x∈ B$  There exist $ϵ&gt;0$ such that $ B(f(x), ϵ) ∩ B(g(x), ϵ)=\varnothing$ for all $ x∈ B$ ${\rm int }(f(B)) ∩ {\rm int }(g(B))=\varnothing$ , where ${\rm int}(E)$ denotes the  interior of a set $E$      How can I solve this problem? Can anyone help?
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