Theorems · Theorem · functional analysis

HasFDerivWithinAt.mapsTo_tangent_cone

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] {F : Type u_3} [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F] {f : E → F}
  {s : Set E} {f' : E →L[𝕜] F} {x : E},
  HasFDerivWithinAt f f' s x → Set.MapsTo (⇑f') (tangentConeAt 𝕜 s x) (tangentConeAt 𝕜 (f '' s) (f x))

The image of a tangent cone under the differential of a map is included in the tangent cone to the image.

Defined in

Mathlib.Analysis.Calculus.FDeriv.Equiv

Cited by

1 results in Mathlib

Foundations

Depth 168 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace