Theorems · Theorem · general topology

TendstoUniformlyOn.continuousOn

∀ {α : Type u_1} {β : Type u_2} {ι : Type u_3} [inst : TopologicalSpace α] [inst_1 : UniformSpace β] {F : ι → α → β}
  {f : α → β} {s : Set α} {p : Filter ι},
  TendstoUniformlyOn F f p s → (∃ᶠ (n : ι) in p, ContinuousOn (F n) s) → ContinuousOn f s

A uniform limit on a set of functions which are continuous on this set is itself continuous on this set.

Defined in

Mathlib.Topology.UniformSpace.UniformApproximation

Cited by

6 results in Mathlib

Foundations

Depth 82 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceUniformSpace