Theorems · Theorem · real analysis

contDiff_iff_iteratedDeriv

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {F : Type u_2} [inst_1 : NormedAddCommGroup F]
  [inst_2 : NormedSpace 𝕜 F] {f : 𝕜 → F} {n : ℕ∞},
  ContDiff 𝕜 (↑n) f ↔
    (∀ (m : ℕ), ↑m ≤ n → Continuous (iteratedDeriv m f)) ∧ ∀ (m : ℕ), ↑m < n → Differentiable 𝕜 (iteratedDeriv m f)

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

Defined in

Mathlib.Analysis.Calculus.IteratedDeriv.Defs

Cited by

4 results in Mathlib

Foundations

Depth 196 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpace