Approach 2 (By Maximum/Minimum principle): Let u,Ω,V
 be as before in approach 1. Let {Si}
 be the set of connected components of Ω
 which intersect with V
. Let Vi=V∩Si
 Then V=⋃iVi⇒u(V)=⋃iu(Vi).

Since each Vi
 is connected, u(Vi)
 is also connected in R
. Thus each u(Vi)
 is an interval in R.
 Observe that u
 is a non-constant harmonic function on open set Vi
 (similar argument as Di
), therefore u
 does not have extremums on each Vi
 by maximum/minimum principles. Thus each u(Vi)
 is an open interval in R
 ⇒u(V)
 is an open set in R
 since arbitrary union of open sets is open.