Theorems · Theorem · measure theory
MeasureTheory.tendsto_zero_of_hasDerivAt_of_integrableOn_Ioi
∀ {E : Type u_1} {f f' : ℝ → E} {a : ℝ} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E],
(∀ x ∈ Set.Ioi a, HasDerivAt f (f' x) x) →
MeasureTheory.IntegrableOn f' (Set.Ioi a) MeasureTheory.volume →
MeasureTheory.IntegrableOn f (Set.Ioi a) MeasureTheory.volume → Filter.Tendsto f Filter.atTop (nhds 0)If a function and its derivative are integrable on (a, +∞), then the function tends to zero
at +∞.
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 269 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
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Cites32
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- Setstatement and proof · cited by 53,352
- Realstatement and proof · cited by 25,697
- RingHom.idproof · cited by 18,349
- NormedAddCommGroupstatement and proof · cited by 15,752
- NormedSpacestatement and proof · cited by 12,499
- ENNRealproof · cited by 9,879
- nhdsstatement and proof · cited by 5,554
- ContinuousLinearMapproof · cited by 5,352
- Filter.Tendstostatement and proof · cited by 3,814
- Filter.atTopstatement and proof · cited by 2,405
- Set.Ioistatement and proof · cited by 1,463
Cited by2
Results whose statement or proof uses this declaration.
- MeasureTheory.integral_eq_zero_of_hasDerivAt_of_integrableproof · cited by 0