continuous_of_locally_uniform_approx_of_continuousAt

English

The indexed product of spaces with ultra-uniformities is ultra-uniform.

Русский

Индексированное произведение пространств с ультра-равномерностями сохраняет ультра-равномерность.

LaTeX

$$$IsUltraUniformity\big(\Pi i, X_i\big)$$$

Lean4

/-- A function which can be locally uniformly approximated by continuous functions is continuous. -/
theorem continuous_of_locally_uniform_approx_of_continuousAt
    (L : ∀ x : α, ∀ u ∈ 𝓤 β, ∃ t ∈ 𝓝 x, ∃ F, ContinuousAt F x ∧ ∀ y ∈ t, (f y, F y) ∈ u) : Continuous f :=
  continuous_iff_continuousAt.2 fun x => continuousAt_of_locally_uniform_approx_of_continuousAt (L x)

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