If each finger height exceeds
half
the distance between the points on
the x-axis, the two Vs will intersect nontrivially, and
otherwise not; this is worth remembering for later.
By the way, when Vε(x) and
Vε'(y) do meet (i.e., when
ε > ½|x-y| < ε' ),
it’s at the point (½(x+y), ½|x-y|) .
 
For our double cover, we look to the connected double cover
of the circle for motivation: