Theorems · Theorem · real analysis

contDiffOn_succ_iff_deriv_of_isOpen

∀ {𝕜 : Type u_1} {F : Type u_2} [inst : NontriviallyNormedField 𝕜] [inst_1 : NormedAddCommGroup F]
  [inst_2 : NormedSpace 𝕜 F] {n : WithTop ℕ∞} {f : 𝕜 → F} {s : Set 𝕜},
  IsOpen s →
    (ContDiffOn 𝕜 (n + 1) f s ↔ DifferentiableOn 𝕜 f s ∧ (n = ⊤ → AnalyticOn 𝕜 f s) ∧ ContDiffOn 𝕜 n (deriv f) s)

A function is C^(n + 1) on an open domain if and only if it is differentiable there, and its derivative (formulated with deriv) is C^n.

Defined in

Mathlib.Analysis.Calculus.ContDiff.Deriv

Cited by

2 results in Mathlib

Foundations

Depth 198 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpace