da Luke1984 » 03/03/2007, 18:47
Let be $a>0$.
Suppose exists a function $f:[0,1]->RR$ such that $|f(x)-f(y)| \geq a$ for every $x ne y$.
Let be $B:=\{I\in P(RR): $ exists $k\in ZZ$ such that $ I=[ka,(k+1)a) \}$.
For every $x\in [0,1]$ there is $I_x in B$ such that $x in I_x$ (because the collection $B$ covers $RR$).
Moreover $I_x$ is unique, because the elements of $B$ are disjoint.
Finally if $x,y in [0,1]$, we have $I_x ne I_y$ (if not we would have $|f(x)-f(y)| < a$).
So there is an injection $g$ from $[0,1]$ to $B$.
On the other hand an injection from $B$ to $ZZ$ obviously exists.
So there is an injection from $[0,1]$ to $ZZ$, and this is absurd.