Theorems · Theorem · global analysis

HasFDerivAt.lim

∀ {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} {f' : E →L[𝕜] F} {x : E} [ContinuousAdd E] [ContinuousSMul 𝕜 E]
  [ContinuousAdd F] [ContinuousSMul 𝕜 F],
  HasFDerivAt f f' x →
    ∀ (v : E) {α : Type u_4} {c : α → 𝕜} {l : Filter α},
      Filter.Tendsto (fun n => ‖c n‖) l Filter.atTop →
        Filter.Tendsto (fun n => c n • (f (x + (c n)⁻¹ • v) - f x)) l (nhds (f' v))

Directional derivative agrees with HasFDeriv.

Defined in

Mathlib.Analysis.Calculus.FDeriv.Basic

Cited by

1 results in Mathlib

Foundations

Depth 167 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldAddCommGroupModuleTopologicalSpaceAddCommGroupModuleTopologicalSpaceContinuousAddContinuousSMulContinuousAddContinuousSMul