If a bounded function $f:\Bbb R\to \Bbb R $ and $\left|\,f(x)-f(y)\right|<\left|x-y\right|$ for $x\ne y$, then there is an $a$ s.t. $f(a)=a$.
What I know is $f$ should be uniformly continuous. As far as I know, for the fixed point theorem to holds, the above condition is not strong enough to imply a fixed point exists as there should be a contraction,i.e.$\left|\,f(x)-f(y)\right|<\alpha\,\left|x-y\right|$ where $\alpha<1$.
This is a question I saw from a book but now I wonder if it is true .