Theorems · Theorem · real analysis

ContDiff.continuous_fderiv_apply

∀ {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}
  {n : WithTop ℕ∞}, ContDiff 𝕜 n f → n ≠ 0 → Continuous fun p => (fderiv 𝕜 f p.1) p.2

If a function is at least C^1, its bundled derivative (mapping (x, v) to Df(x) v) is continuous.

Defined in

Mathlib.Analysis.Calculus.ContDiff.Defs

Cited by

0 results in Mathlib

Foundations

Depth 200 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace