Theorems · Theorem · general topology

ContinuousOn.strictMonoOn_of_injOn_Ioo

∀ {α : Type u} [inst : TopologicalSpace α] [inst_1 : ConditionallyCompleteLinearOrder α] [OrderTopology α]
  [DenselyOrdered α] {δ : Type u_1} [inst_4 : LinearOrder δ] [inst_5 : TopologicalSpace δ] [OrderClosedTopology δ]
  {a b : α} {f : α → δ},
  a < b →
    ContinuousOn f (Set.Ioo a b) →
      Set.InjOn f (Set.Ioo a b) → StrictMonoOn f (Set.Ioo a b) ∨ StrictAntiOn f (Set.Ioo a b)

Every continuous injective f : (a, b) → δ is strictly monotone or antitone (increasing or decreasing).

Defined in

Mathlib.Topology.Order.IntermediateValue

Cited by

0 results in Mathlib

Foundations

Depth 95 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceConditionallyCompleteLinearOrderOrderTopologyDenselyOrderedLinearOrderTopologicalSpaceOrderClosedTopology