Theorems · Theorem · real analysis
contDiffOn_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), ↑m ≤ n → ContinuousOn (iteratedDerivWithin m f s) s) ∧
∀ (m : ℕ), ↑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.
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 187 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites14
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement and proof · cited by 53,352
- NormedAddCommGroupstatement and proof · cited by 15,752
- NormedSpacestatement and proof · cited by 12,499
- NontriviallyNormedFieldstatement and proof · cited by 8,742
- ENatstatement and proof · cited by 4,985
- ContinuousOnstatement and proof · cited by 1,411
- WithTop.somestatement · cited by 1,128
- DifferentiableOnstatement and proof · cited by 419
- ContDiffOnstatement · cited by 294
- UniqueDiffOnstatement and proof · cited by 215
- iteratedDerivWithinstatement and proof · cited by 122
- ContinuousMultilinearMap.piFieldEquivproof · cited by 15
Cited by1
Results whose statement or proof uses this declaration.
- contDiffOn_nat_iff_continuousOn_differentiableOn_derivproof · cited by 1