Theorems · Theorem · general topology

MonotoneOn.countable_not_continuousWithinAt

∀ {α : Type u_1} {β : Type u_2} [inst : LinearOrder α] [inst_1 : TopologicalSpace α] [OrderTopology α]
  [inst_3 : LinearOrder β] {s : Set α} {f : α → β} [inst_4 : TopologicalSpace β] [OrderTopology β]
  [SecondCountableTopology β], MonotoneOn f s → {x | x ∈ s ∧ ¬ContinuousWithinAt f s x}.Countable

In a second countable space, the set of points where a monotone function is not continuous within a set is at most countable.

Defined in

Mathlib.Topology.Order.Monotone

Cited by

3 results in Mathlib

Foundations

Depth 95 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

LinearOrderTopologicalSpaceOrderTopologyLinearOrderTopologicalSpaceOrderTopologySecondCountableTopology