Theorems · Theorem · general topology

TendstoLocallyUniformly.continuous

∀ {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} {p : Filter ι}, TendstoLocallyUniformly F f p → (∃ᶠ (n : ι) in p, Continuous (F n)) → Continuous f

A locally uniform limit of continuous functions is continuous.

Defined in

Mathlib.Topology.UniformSpace.UniformApproximation

Cited by

1 results in Mathlib

Foundations

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

Assumes

TopologicalSpaceUniformSpace