continuousAt_iff_isLittleO — Mathlib

English

The identity map is little-o of itself at 0, i.e., (id) = o_{nhds 0} (id).

Русский

Идентификатор есть little-o относительно самого себя в точке 0.

LaTeX

$$$ (\lambda x: x) =o[\mathcal{N}0] (\text{id}). $$$

Lean4

theorem continuousAt_iff_isLittleO {α : Type*} {E : Type*} [NormedRing E] [NormOneClass E] [TopologicalSpace α]
    {f : α → E} {x : α} : (ContinuousAt f x) ↔ (fun (y : α) ↦ f y - f x) =o[𝓝 x] (fun (_ : α) ↦ (1 : E)) := by
  simp [ContinuousAt, ← tendsto_sub_nhds_zero_iff]

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