∀ {α : Type u_2} {β : Type u_3} [inst : LinearOrder α] [inst_1 : TopologicalSpace α] [inst_2 : TopologicalSpace β]
  [ClosedIciTopology α] [Nonempty β] [inst_5 : One α] {f : β → α},
  Continuous f → HasCompactMulSupport f → ∃ x, ∀ (y : β), f y ≤ f x