Theorems · Theorem · general topology

ContinuousOn.union_continuousAt

∀ {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst_1 : TopologicalSpace β] {s t : Set α} {f : α → β},
  IsOpen s → ContinuousOn f s → (∀ x ∈ t, ContinuousAt f x) → ContinuousOn f (s ∪ t)

If f is continuous on an open set s and continuous at each point of another set t then f is continuous on s ∪ t.

Defined in

Mathlib.Topology.ContinuousOn

Cited by

1 results in Mathlib

Foundations

Depth 67 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceTopologicalSpace