differentiableWithinAt_Ioi_iff_Ici

English

The equivalence for whether derivWithin f t x lies in a set s is given by a standard two-case condition depending on differentiability at x.

Русский

Эквиваленция для того, лежит ли derivWithin f t x в s, задаётся двумя случаями в зависимости от дифференцируемости в x.

LaTeX

$$$$ \\ derivWithin f t x ∈ s \\iff ( \\text{DifferentiableWithinAt } 𝕜 f t x ∧ derivWithin f t x ∈ s) ∨ (¬\\text{DifferentiableWithinAt } 𝕜 f t x ∧ 0 ∈ s) $$$$

Lean4

theorem differentiableWithinAt_Ioi_iff_Ici [PartialOrder 𝕜] :
    DifferentiableWithinAt 𝕜 f (Ioi x) x ↔ DifferentiableWithinAt 𝕜 f (Ici x) x :=
  ⟨fun h => h.hasDerivWithinAt.Ici_of_Ioi.differentiableWithinAt, fun h =>
    h.hasDerivWithinAt.Ioi_of_Ici.differentiableWithinAt⟩
    -- Golfed while splitting the file

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