Let $B^-$ be the open unit ball in $\mathbb{R}^n$ for $n\in\mathbb{N}$, and let $f$ be a uniformly continuous function from $B^-$ to itself, with a uniformly continuous inverse $f^{-1}$.
Under these conditions, must $f$ have a fixed point in $B^-$?
$\begingroup$
$\endgroup$
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
2
No, for instance a piecewise-linear (say with two pieces) bijection $f:(0,1)\to(0,1)$ whose graph lies on one side of the diagonal $y=x$.
-
$\begingroup$ thank you! I must admit I thought it was harder...but it's also very nice to get a short and clear answer. Cheers, Frank (I understand the vote to close, by the way...but that's how it sometimes goes) $\endgroup$ Commented May 6, 2018 at 16:07
-
$\begingroup$ So now I'll start thinking again about which natural extra conditions on $f$ would ensure a fixed point in the open ball... $\endgroup$ Commented May 6, 2018 at 16:08