Theorems · Theorem · general topology

ContinuousOn.image_Icc_of_monotoneOn

∀ {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.Icc a b) → MonotoneOn f (Set.Icc a b) → f '' Set.Icc a b = Set.Icc (f a) (f b)

Defined in

Mathlib.Topology.Order.IntermediateValue

Cited by

5 results in Mathlib

Foundations

Depth 87 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceConditionallyCompleteLinearOrderOrderTopologyDenselyOrderedLinearOrderTopologicalSpaceOrderClosedTopology