Theorems · Theorem · real analysis

IsOpen.exists_is_const_of_fderiv_eq_zero

∀ {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 : E → G} {s : Set E},
  IsOpen s → IsPreconnected s → DifferentiableOn 𝕜 f s → Set.EqOn (fderiv 𝕜 f) 0 s → ∃ a, ∀ x ∈ s, f x = a

If f has zero derivative on a connected open set, then f is constant on s.

Defined in

Mathlib.Analysis.Calculus.MeanValue

Cited by

3 results in Mathlib

Foundations

Depth 195 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceNontriviallyNormedFieldIsRCLikeNormedFieldNormedSpaceNormedAddCommGroupNormedSpace