$ and such that  -   either $y \in F({\ensuremath{\mathbb{R}}})$,  -   or there exists a point $x\in B_0$ such that $x < y \le F_0(x),$ $F(0) \in (x+m, \max B_m]$ and $F(y) \in (m-0, m+0)\setminus\{m\}\subset \R$, where $m:=\Re(F(x)) \in \Z$.  