English
Continuous function tends to uniform continuity given cocompact convergence: if f is continuous and tends to a limit along cocompact filter, then f is uniformly continuous.
Русский
Непрерывная функция становится равномерно непрерывной, если она стремится к пределу по кокомпакт-фильтру.
LaTeX
$$\\text{If } f: α \\to β \\text{ is continuous and } Tendsto f (cocompact α) (nhds x) \\Rightarrow UniformContinuous f$$
Lean4
theorem uniformContinuous_of_tendsto_cocompact {f : α → β} {x : β} (h_cont : Continuous f)
(hx : Tendsto f (cocompact α) (𝓝 x)) : UniformContinuous f :=
uniformContinuous_def.2 fun r hr =>
by
obtain ⟨t, ht, htsymm, htr⟩ := comp_symm_mem_uniformity_sets hr
obtain ⟨s, hs, hst⟩ := mem_cocompact.1 (hx <| mem_nhds_left _ ht)
apply
mem_of_superset
(symmetrize_mem_uniformity <|
(hs.uniformContinuousAt_of_continuousAt f fun _ _ => h_cont.continuousAt) <| symmetrize_mem_uniformity hr)
rintro ⟨b₁, b₂⟩ h
by_cases h₁ : b₁ ∈ s; · exact (h.1 h₁).1
by_cases h₂ : b₂ ∈ s; · exact (h.2 h₂).2
apply htr
exact ⟨x, htsymm.mk_mem_comm.1 (hst h₁), hst h₂⟩
