∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℂ E] {U : Set E} {g : E → ℂ},
  AnalyticOnNhd ℂ g U → IsPreconnected U → (∃ w, ∀ z ∈ U, g z = w) ∨ ∀ s ⊆ U, IsOpen s → IsOpen (g '' s)