Theorems · Theorem · general topology

continuousOn_iUnion_iff_of_isOpen

∀ {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst_1 : TopologicalSpace β] {f : α → β} {ι : Type u_5}
  {s : ι → Set α}, (∀ (i : ι), IsOpen (s i)) → (ContinuousOn f (⋃ i, s i) ↔ ∀ (i : ι), ContinuousOn f (s i))

A function is continuous on a union of open sets s i iff it is continuous on each s i.

Defined in

Mathlib.Topology.ContinuousOn

Cited by

0 results in Mathlib

Foundations

Depth 69 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceTopologicalSpace