Theorems · Theorem · general topology

intermediate_value_Icc

• #79 of the 100 theorems: The Intermediate Value Theorem
• 1000+ list: Intermediate value theorem

∀ {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}, a ≤ b → ∀ {f : α → δ}, ContinuousOn f (Set.Icc a b) → Set.Icc (f a) (f b) ⊆ f '' Set.Icc a b

Intermediate Value Theorem for continuous functions on closed intervals, case f a ≤ t ≤ f b.

Defined in

Mathlib.Topology.Order.IntermediateValue

Cited by

4 results in Mathlib

Foundations

Depth 86 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceConditionallyCompleteLinearOrderOrderTopologyDenselyOrderedLinearOrderTopologicalSpaceOrderClosedTopology