Theorems · Theorem · real analysis

ContDiffAt.differentiableAt

∀ {𝕜 : Type u} [inst : NontriviallyNormedField 𝕜] {E : Type uE} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] {F : Type uF} [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F] {f : E → F}
  {x : E} {n : WithTop ℕ∞}, ContDiffAt 𝕜 n f x → n ≠ 0 → DifferentiableAt 𝕜 f x

If a function is C^n with n ≥ 1 at a point, then it is differentiable there.

Defined in

Mathlib.Analysis.Calculus.ContDiff.Defs

Cited by

14 results in Mathlib

Foundations

Depth 184 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace