Theorems · Theorem · general topology

intermediate_value_uIcc

∀ {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}, ContinuousOn f (Set.uIcc a b) → Set.uIcc (f a) (f b) ⊆ f '' Set.uIcc a b

Intermediate Value Theorem for continuous functions on closed intervals, unordered case.

Defined in

Mathlib.Topology.Order.IntermediateValue

Cited by

3 results in Mathlib

Foundations

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

Assumes

TopologicalSpaceConditionallyCompleteLinearOrderOrderTopologyDenselyOrderedLinearOrderTopologicalSpaceOrderClosedTopology