Theorems · Theorem · general topology

Continuous.strictMonoOn_of_inj_rigidity

∀ {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]
  {f : α → δ}, Continuous f → Function.Injective f → ∀ {a b : α}, a < b → StrictMonoOn f (Set.Icc a b) → StrictMono f

Suppose α is equipped with a conditionally complete linear dense order and f : α → δ is continuous and injective. Then f is strictly monotone (increasing) if it is strictly monotone (increasing) on some closed interval [a, b].

Defined in

Mathlib.Topology.Order.IntermediateValue

Cited by

1 results in Mathlib

Foundations

Depth 93 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceConditionallyCompleteLinearOrderOrderTopologyDenselyOrderedLinearOrderTopologicalSpaceOrderClosedTopology