Theorems · Theorem · general topology

IsPreconnected.intermediate_value

∀ {X : Type u} {α : Type v} [inst : TopologicalSpace X] [inst_1 : LinearOrder α] [inst_2 : TopologicalSpace α]
  [OrderClosedTopology α] {s : Set X},
  IsPreconnected s → ∀ {a b : X}, a ∈ s → b ∈ s → ∀ {f : X → α}, ContinuousOn f s → Set.Icc (f a) (f b) ⊆ f '' s

Intermediate Value Theorem for continuous functions on connected sets.

Defined in

Mathlib.Topology.Order.IntermediateValue

Cited by

6 results in Mathlib

Foundations

Depth 81 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceLinearOrderTopologicalSpaceOrderClosedTopology