Theorems · Theorem · real analysis

differentiableWithinAt_of_fderivWithin_injective

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : AddCommGroup E] [inst_2 : Module 𝕜 E]
  [inst_3 : TopologicalSpace E] {F : Type u_3} [inst_4 : AddCommGroup F] [inst_5 : Module 𝕜 F]
  [inst_6 : TopologicalSpace F] {f : E → F} {x : E} {s : Set E},
  Function.Injective ⇑(fderivWithin 𝕜 f s x) → DifferentiableWithinAt 𝕜 f s x

If f : E → F has injective differential within s at x, it is differentiable within s at x.

Defined in

Mathlib.Analysis.Calculus.FDeriv.Const

Cited by

2 results in Mathlib

Foundations

Depth 174 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldAddCommGroupModuleTopologicalSpaceAddCommGroupModuleTopologicalSpace