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Let $f:\left\{x\in\mathbb{R}^n\vert\parallel x\parallel<1\right\}\rightarrow\mathbb{R}$ be an one-to-one bounded continuous function.
I want to construct such $f$ which is not uniformly continuous.

In this case, I thought I can construct $f$ with a restriction $n=2$.
But I'm confused because $f$ is bounded so I can't use functions like $\frac{1}{x}$ on $(0,1)$.
To top that off, $f$ is even one-to-one so I gave up and now I'm writing this to get some help from you who is smarter than me.

Please give me some help to solve this problem.
Thanks.

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For $n=1$, as pointed out by Vishal, there is no such function $f$.

For $n\geq 2$, such an $f$ doesn't exist either, but for a "trivial" reason: there is no continuous and one-to-one function at all from the open unit ball $B\subset\mathbb R^n$ into $\mathbb R$. One can see this as follows. Assume that $f:B\to\mathbb R$ is continuous and $1$-$1$. Then $I=f(B)$ is a nontrivial interval of $\mathbb R$ because $B$ is connected and $f$ is continuous and non-constant. Take any point $a\in B$ such that $f(a)$ is an interior point of $I$. Then $f(B\setminus\{ a\})$ has to be connected because $B\setminus\{ a\}$ is connected ($n\geq 2$) and $f$ is continuous. But $f(B\setminus\{ a\})$ is equal to $I\setminus \{ f(a)\}$ because $f$ is $1$-$1$, and this is not a connected set. So we have a contradiction.

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At least for $n =1$, any such function will have a limit on the end points $-1,1$ in this case and hence will have an extension on the interval $[-1,1]$ and hence will be uniformly continuous. So, you cannot construct a function you want in the case $n=1$.

For higher dimensions, the main thing is to examine whether the limit exists on the boundary of the closed ball (in the case $n=1$ one-one and continuity implies monotonicity, which gives the limit). If there is a limit, then the function will again be uniformly continuous.

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You can't construct such function. You can prove that $f$ has the limit on the boundary of the closed ball and $f$ will be uniformly continuous.

Let $U_n$ be the open unit ball in $\mathbb{R}^n$. Consider $f$ on the interval $[0,x)\subset U_n$ that connects the center of the ball and the point $x$ from the edge of the ball (i.e. $\|x\|=1$). Now we consider $f$ only $[0,x)$ and using that we can consider $f$ as a function of one variable: $f(y)=f(tx), t\in[0,1)$. Since $f$ is continuous on $[0,x)$ and one-to-one then it is monotonic and hence has the limit at $x$.

Actually, I must take any curve $C$, that connects the center of the ball and the point $x$, instead of $[0,x)$. But that case is proven similarly.

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