Theorems · Theorem · real analysis

ContDiffPointwiseHolderAt.comp_of_differentiableAt

∀ {E : Type u_1} {F : Type u_2} {G : Type u_3} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E]
  [inst_2 : NormedAddCommGroup F] [inst_3 : NormedSpace ℝ F] [inst_4 : NormedAddCommGroup G] [inst_5 : NormedSpace ℝ G]
  {k : ℕ} {α : ↑unitInterval} {f : E → F} (a : E) {g : F → G},
  ContDiffPointwiseHolderAt k α g (f a) →
    ContDiffPointwiseHolderAt k α f a →
      DifferentiableAt ℝ g (f a) ∨ DifferentiableAt ℝ f a → ContDiffPointwiseHolderAt k α (g ∘ f) a

Composition of two $C^{k+(α)}$ functions is a $C^{k+(α)}$ function, provided that one of them is differentiable. The latter condition follows automatically from the functions being $C^{k+(α)}$, if k ≠ 0, see comp below.

Defined in

Mathlib.Analysis.Calculus.ContDiffHolder.Pointwise

Cited by

3 results in Mathlib

Foundations

Depth 209 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace