continuousOn_of_uniform_approx_of_continuousOn

English

The iInf of a family of uniform spaces, each ultra-uniform, is ultra-uniform.

Русский

Локальное пересечение семейства пространств, каждое из которых ультра-равномерно, образует ультра-равномерность.

LaTeX

$$$@IsUltraUniformity\_ {X} (\big( \bigwedge i U_i \big) )$$$

Lean4

/-- A function which can be uniformly approximated by functions which are continuous on a set
is continuous on this set. -/
theorem continuousOn_of_uniform_approx_of_continuousOn
    (L : ∀ u ∈ 𝓤 β, ∃ F, ContinuousOn F s ∧ ∀ y ∈ s, (f y, F y) ∈ u) : ContinuousOn f s :=
  continuousOn_of_locally_uniform_approx_of_continuousWithinAt fun _x hx u hu =>
    ⟨s, self_mem_nhdsWithin, (L u hu).imp fun _F hF => ⟨hF.1.continuousWithinAt hx, hF.2⟩⟩

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