Theorems · Theorem · real analysis

derivWithin_zero_of_frequently_const

∀ {𝕜 : Type u} [inst : NontriviallyNormedField 𝕜] {F : Type v} [inst_1 : NormedAddCommGroup F]
  [inst_2 : NormedSpace 𝕜 F] {f : 𝕜 → F} {s : Set 𝕜} {x : 𝕜} {c : F},
  (∃ᶠ (y : 𝕜) in nhdsWithin x (s \ {x}), f y = c) → derivWithin f s x = 0

If a function is equal to a constant at a set of points that accumulates to x in s, then its derivative within s at x equals zero, either because it has derivative zero or because it isn't differentiable at this point.

Defined in

Mathlib.Analysis.Calculus.Deriv.Inverse

Cited by

1 results in Mathlib

Foundations

Depth 171 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpace