Is the following statement is True/false ? reagarding closed ball

Question

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 $ϵ>0$ 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

Answer 1

Your argument is not correct. If $0<\epsilon <1$ there cannot be any point common to the two balls. To give a counterexample take $f(x)=(1-\|x\|,0,..,0)$ and $g(x)=(0,0,..,0)$ for all $x \in B$. Then $f(x) \neq g(x)$ for all $x \in B$ and there is no $\epsilon$ such that $B(f(x),\epsilon)\cap B(g(x),\epsilon)=\emptyset$ for all $x \in B$. To see this take $x =(1-\epsilon /2,0,0...,0)$ and note that $(0,\cdots,0)$ is a point in the intersection of the two balls.

Answer 2

$f(x) \ne g(x)$ for any $x\in B$. So $d(f(x), g(x)) > 0$ for all $x \in B$. So $A= \{d(f(x),g(x))|x\in B\}$ is non empty and bounded below by $0$ so $\inf A$ exist.

If $\inf A > 0$ then for any $0< \epsilon < \frac 12\inf A$. We'd be done. The statement is true[1].

But what if $\inf A = 0$. Then the statement is false[2].

So we can surely find some acceptable $f$ and $g$ where $\inf \{d(f(x),g(x))|x\in B\} = 0$.[3]

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1)

If $c \in B(f(x),\epsilon)\cap B(g(x),\epsilon)$ then by triangle inequality $d(f(x),g(x)) \le d(f(x),c) + d(g(c),c) < \epsilon + \epsilon < d(f(x),g(x))$ that 's a contradiction.

2)

For any $\epsilon > 0$ there exists an $x$ where $d(f(x), g(x)) < \epsilon$ so $g(x) \in B(f(x), \epsilon)$ and $f(x) \in B(g(x),\epsilon)$ and $g(x) \in B(g(x),\epsilon)$ and $f(x) \in B(f(x), \epsilon)$ so $f(x), g(x) \in B(g(x),\epsilon)\cap B(f(x), \epsilon)$.

3)

Let $f(x)=x$ be the identity function let $g(x)= (2,0,0,0,0,....) - x$. Let for any $1 > \delta > 0$ then $z= (1-\frac 12\delta, 0,0,0,.....) \in B$ and $d(f(z), g(z)) = d((1-\frac 12\delta,0,0,0....) ,(1+\frac 12\delta,0,0,0,....)) = \delta$. So for all $\delta > 0$ there is a $z \in B$ so that $d(f(x), g(x)) = \delta$ so $\inf \{d(f(x),g(x))|x\in B\} = 0$.