Theorems · Theorem · real analysis
hasDerivWithinAt_iff_tendsto_slope
∀ {𝕜 : Type u} [inst : NontriviallyNormedField 𝕜] {F : Type v} [inst_1 : NormedAddCommGroup F]
[inst_2 : NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜} {s : Set 𝕜},
HasDerivWithinAt f f' s x ↔ Filter.Tendsto (slope f x) (nhdsWithin x (s \ {x})) (nhds f')- Defined in
- Mathlib.Analysis.Calculus.Deriv.Slope
- Cited by
- 6 results in Mathlib
- Foundations
- Depth 168 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites16
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement and proof · cited by 53,352
- NormedAddCommGroupstatement and proof · cited by 15,752
- NormedSpacestatement and proof · cited by 12,499
- NontriviallyNormedFieldstatement and proof · cited by 8,742
- nhdsstatement and proof · cited by 5,554
- Filter.Tendstostatement and proof · cited by 3,814
- Compl.complproof · cited by 2,925
- nhdsWithinstatement · cited by 1,912
- SProd.sprodproof · cited by 1,750
- Filter.principalproof · cited by 740
- HasDerivWithinAtstatement · cited by 333
- slopestatement and proof · cited by 147
Cited by6
Results whose statement or proof uses this declaration.
- hasDerivWithinAt_iff_tendsto_slope'proof · cited by 4
- HasDerivWithinAt.nonneg_of_monotoneOnproof · cited by 4
- range_derivWithin_subset_closure_span_imageproof · cited by 3
- HasDerivWithinAt.limsup_norm_slope_leproof · cited by 2
- not_differentiableWithinAt_of_deriv_tendsto_atTop_Ioiproof · cited by 2
- HasDerivWithinAt.limsup_slope_leproof · cited by 0