∀ {α : Type u_1} {β : Type u_2} [inst : LinearOrder α] [inst_1 : TopologicalSpace α] [OrderTopology α]
  [inst_3 : LinearOrder β] [inst_4 : TopologicalSpace β] [OrderTopology β] {f : α → β} {s : Set α} {a : α},
  MonotoneOn f s →
    s ∈ nhds a →
      (∀ b < f a, ∃ c ∈ s, f c ∈ Set.Ioo b (f a)) → (∀ b > f a, ∃ c ∈ s, f c ∈ Set.Ioo (f a) b) → ContinuousAt f a