Theorems · Theorem · real analysis

contDiffWithinAt_iff_contDiffOn_nhds

∀ {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] {s : Set E}
  {f : E → F} {x : E} {n : WithTop ℕ∞},
  n ≠ ↑⊤ → (ContDiffWithinAt 𝕜 n f s x ↔ ∃ u ∈ nhdsWithin x (insert x s), ContDiffOn 𝕜 n f u)

A function is C^n within a set at a point, for n : ℕ, if and only if it is C^n on a neighborhood of this point.

Defined in

Mathlib.Analysis.Calculus.ContDiff.Defs

Cited by

0 results in Mathlib

Foundations

Depth 179 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace