continuousAt_length — Mathlib

English

For every list l, the function List.length is continuous at l.

Русский

Для каждого списка l функция длины списка непрерывна в точке l.

LaTeX

$$$ \operatorname{ContinuousAt}(\text{List.length})(l) $$$

Lean4

theorem continuousAt_length : ∀ l : List α, ContinuousAt List.length l :=
  by
  simp only [ContinuousAt, nhds_discrete]
  refine tendsto_nhds ?_ ?_
  · exact tendsto_pure_pure _ _
  · intro l a ih
    dsimp only [List.length]
    refine Tendsto.comp (tendsto_pure_pure (fun x => x + 1) _) ?_
    exact Tendsto.comp ih tendsto_snd

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