Theorems · Theorem · general topology

IsCompact.exists_isMaxOn

∀ {α : Type u_2} {β : Type u_3} [inst : LinearOrder α] [inst_1 : TopologicalSpace α] [inst_2 : TopologicalSpace β]
  [ClosedIciTopology α] {s : Set β}, IsCompact s → s.Nonempty → ∀ {f : β → α}, ContinuousOn f s → ∃ x ∈ s, IsMaxOn f s x

The extreme value theorem: a continuous function realizes its maximum on a compact set.

Defined in

Mathlib.Topology.Order.Compact

Cited by

22 results in Mathlib

Foundations

Depth 80 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

LinearOrderTopologicalSpaceTopologicalSpaceClosedIciTopology