unique_on — Mathlib

English

HasFDerivAtFilter is equivalent to a limit statement that expresses the derivative via a normalized error term tending to zero.

Русский

Эквивалентность производной и предела через нормированную ошибку, tending to ноль.

LaTeX

$$$\text{HasFDerivAtFilter } f f' x L \iff \lim_{L} \frac{\|f(y)-f(x)-f'(y-x)\|}{\|y-x\|} = 0$$$

Lean4

/-- If `f'` and `f₁'` are two derivatives of `f` within `s` at `x`, then they are equal on the
tangent cone to `s` at `x` -/
theorem unique_on (hf : HasFDerivWithinAt f f' s x) (hg : HasFDerivWithinAt f f₁' s x) :
    EqOn f' f₁' (tangentConeAt 𝕜 s x) := fun _ ⟨_, _, dtop, clim, cdlim⟩ =>
  tendsto_nhds_unique (hf.lim atTop dtop clim cdlim) (hg.lim atTop dtop clim cdlim)

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