continuousOn — Mathlib

English

An iInf of a family of ultra-uniformities is ultra-uniform.

Русский

iInf семейства ультра-равномерностей является ультра-равномерным.

LaTeX

$$$IsUltraUniformity\big(\bigsqcap_i U_i\big)$$$

Lean4

/-- A locally uniform limit on a set of functions which are continuous on this set is itself
continuous on this set. -/
protected theorem continuousOn (h : TendstoLocallyUniformlyOn F f p s) (hc : ∀ᶠ n in p, ContinuousOn (F n) s)
    [NeBot p] : ContinuousOn f s :=
  by
  refine continuousOn_of_locally_uniform_approx_of_continuousWithinAt fun x hx u hu => ?_
  rcases h u hu x hx with ⟨t, ht, H⟩
  rcases (hc.and H).exists with ⟨n, hFc, hF⟩
  exact ⟨t, ht, ⟨F n, hFc.continuousWithinAt hx, hF⟩⟩

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