fderivWithin_right — Mathlib

English

There exists a function f' giving the derivative with respect to y that is ContDiffWithinAt.

Русский

Существует функция f', задающая производную по y, которая гладкая внутри заданного множества.

LaTeX

$$$\exists f': E \to F \toL[\mathbb{K}] G,\; (\forall x \in v, HasFDerivWithinAt (f x) (f' x) t (g x))$$$

Lean4

/-- `fderivWithin 𝕜 f s` is smooth at `x₀` within `s`. -/
theorem fderivWithin_right (hf : ContDiffWithinAt 𝕜 n f s x₀) (hs : UniqueDiffOn 𝕜 s) (hmn : m + 1 ≤ n)
    (hx₀s : x₀ ∈ s) : ContDiffWithinAt 𝕜 m (fderivWithin 𝕜 f s) s x₀ :=
  ContDiffWithinAt.fderivWithin (ContDiffWithinAt.comp (x₀, x₀) hf contDiffWithinAt_snd <| prod_subset_preimage_snd s s)
    contDiffWithinAt_id hs hmn hx₀s (by rw [preimage_id'])

Похожие утверждения