In such a V_i, n we can therefore find two x-axis points that are closer together than 2/n , and for such points, their basic open Vs of finger-height 1/n ... intersect nontrivially, and the presumed-continuous section f is forced to send that point of intersection to both the upper and the lower half-planes in the double-cover, a contradiction.
  So no section for p can be continuous. Of interest to topos theorists ...                   [ >> ]