Theorems · Theorem · real analysis

IsOpen.eqOn_of_fderiv_eq

∀ {E : Type u_1} [inst : NormedAddCommGroup E] [NormedSpace ℝ E] {𝕜 : Type u_3} {G : Type u_4}
  [inst_2 : NontriviallyNormedField 𝕜] [IsRCLikeNormedField 𝕜] [inst_4 : NormedSpace 𝕜 E]
  [inst_5 : NormedAddCommGroup G] [inst_6 : NormedSpace 𝕜 G] {f g : E → G} {s : Set E} {x : E},
  IsOpen s →
    IsPreconnected s →
      DifferentiableOn 𝕜 f s →
        DifferentiableOn 𝕜 g s → (∀ x ∈ s, fderiv 𝕜 f x = fderiv 𝕜 g x) → x ∈ s → f x = g x → Set.EqOn f g s

If two functions have equal Fréchet derivatives at every point of a connected open set, and are equal at one point in that set, then they are equal on that set.

Defined in

Mathlib.Analysis.Calculus.MeanValue

Cited by

1 results in Mathlib

Foundations

Depth 197 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceNontriviallyNormedFieldIsRCLikeNormedFieldNormedSpaceNormedAddCommGroupNormedSpace