Theorems · Theorem · real analysis
derivWithin_mem_iff
∀ {𝕜 : Type u} [inst : NontriviallyNormedField 𝕜] {F : Type v} [inst_1 : NormedAddCommGroup F]
[inst_2 : NormedSpace 𝕜 F] {f : 𝕜 → F} {t : Set 𝕜} {s : Set F} {x : 𝕜},
derivWithin f t x ∈ s ↔
DifferentiableWithinAt 𝕜 f t x ∧ derivWithin f t x ∈ s ∨ ¬DifferentiableWithinAt 𝕜 f t x ∧ 0 ∈ s- Defined in
- Mathlib.Analysis.Calculus.Deriv.Basic
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 106 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites7
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
- DifferentiableWithinAtstatement and proof · cited by 453
- derivWithinstatement and proof · cited by 258
- derivWithin_zero_of_not_differentiableWithinAtproof · cited by 12
Cited by1
Results whose statement or proof uses this declaration.
- measurable_derivWithin_Iciproof · cited by 3