hasStrictDerivAt_of_hasDerivAt_of_continuousAt

English

If derivative is bounded by C on a convex interval in 𝕜, then LipschitzOnWith C f on that interval.

Русский

Если производная ограничена на интервале, то функция липшицева на этом интервале.

LaTeX

$$$$ \text{Lipschitz on interval } f \text{ with constant } C $$$$

Lean4

/-- Over the reals or the complexes, a continuously differentiable function is strictly
differentiable. -/
theorem hasStrictDerivAt_of_hasDerivAt_of_continuousAt {f f' : 𝕜 → G} {x : 𝕜}
    (hder : ∀ᶠ y in 𝓝 x, HasDerivAt f (f' y) y) (hcont : ContinuousAt f' x) : HasStrictDerivAt f (f' x) x :=
  hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt (hder.mono fun _ hy => hy.hasFDerivAt) <|
    (smulRightL 𝕜 𝕜 G 1).continuous.continuousAt.comp hcont

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