hasMFDerivAt — Mathlib

English

If s has unique differentiable interior at x and f is differentiable at x, then mfderivWithin equals mfderiv.

Русский

Если вокруг x в s уникально различимо, и f дифференцируема в x, то mfderivWithin равен mfderiv.

LaTeX

$$$ (hs : UniqueMDiffWithinAt I s x) (h : MDifferentiableAt I I' f x) \Rightarrow mfderivWithin I I' f s x = mfderiv I I' f x $$$

Lean4

theorem hasMFDerivAt (h : MDifferentiableAt I I' f x) : HasMFDerivAt I I' f x (mfderiv I I' f x) :=
  by
  refine ⟨h.continuousAt, ?_⟩
  simp only [mfderiv, h, if_pos, mfld_simps]
  exact DifferentiableWithinAt.hasFDerivWithinAt h.differentiableWithinAt_writtenInExtChartAt

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