Theorems · Theorem · real analysis

contDiffOn_nat_iff_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},
  UniqueDiffOn 𝕜 s →
    (ContDiffOn 𝕜 (↑n) f s ↔
      (∀ m ≤ n, ContinuousOn (iteratedDerivWithin m f s) s) ∧ ∀ m < n, DifferentiableOn 𝕜 (iteratedDerivWithin m f s) s)

The property of being C^n, initially defined in terms of the Fréchet derivative, can be reformulated in terms of the one-dimensional derivative on sets with unique derivatives.

Defined in

Mathlib.Analysis.Calculus.IteratedDeriv.Defs

Cited by

1 results in Mathlib

Foundations

Depth 188 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpace