English
The source of a restricted homeomorphism inherits ContDiffOn property from the original source.
Русский
Область определения ограниченного частного гомеоморфа сохраняет свойство ContDiffOn.
LaTeX
$$$\text{ContDiffOn } 𝕜 n f (f.restrContDiff 𝕜 n hn).source$$$
Lean4
/-- The iterated derivative of the sum of two functions is the sum of the iterated derivatives.
This is the same as `iteratedFDerivWithin_add_apply`, but using the spelling `(fun x ↦ f x + g x)`
instead of `f + g`, which can be handy for some rewrites.
TODO: use one form consistently. -/
theorem iteratedFDerivWithin_add_apply' {f g : E → F} (hf : ContDiffWithinAt 𝕜 i f s x)
(hg : ContDiffWithinAt 𝕜 i g s x) (hu : UniqueDiffOn 𝕜 s) (hx : x ∈ s) :
iteratedFDerivWithin 𝕜 i (fun x => f x + g x) s x =
iteratedFDerivWithin 𝕜 i f s x + iteratedFDerivWithin 𝕜 i g s x :=
iteratedFDerivWithin_add_apply hf hg hu hx
