Theorems · Theorem · real analysis

contDiffOn_of_continuousOn_differentiableOn_deriv

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {F : Type u_2} [inst_1 : NormedAddCommGroup F]
  [inst_2 : NormedSpace 𝕜 F] {f : 𝕜 → F} {s : Set 𝕜} {n : ℕ∞},
  (∀ (m : ℕ), ↑m ≤ n → ContinuousOn (fun x => iteratedDerivWithin m f s x) s) →
    (∀ (m : ℕ), ↑m < n → DifferentiableOn 𝕜 (fun x => iteratedDerivWithin m f s x) s) → ContDiffOn 𝕜 (↑n) f s

If the first n derivatives within a set of a function are continuous, and its first n-1 derivatives are differentiable, then the function is C^n. This is not an equivalence in general, but this is an equivalence when the set has unique derivatives, see contDiffOn_iff_continuousOn_differentiableOn_deriv.

Defined in

Mathlib.Analysis.Calculus.IteratedDeriv.Defs

Cited by

0 results in Mathlib

Foundations

Depth 178 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpace