Theorems · Theorem · real analysis
AbsolutelyContinuousOnInterval.continuousOn
∀ {F : Type u_2} [inst : SeminormedAddCommGroup F] {a b : ℝ} {f : ℝ → F},
AbsolutelyContinuousOnInterval f a b → ContinuousOn f (Set.uIcc a b)If f is absolutely continuous on uIcc a b, then f is continuous on uIcc a b.
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 119 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- SeminormedAddCommGroup
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.
- Realstatement and proof · cited by 25,697
- SeminormedAddCommGroupstatement and proof · cited by 2,671
- ContinuousOnstatement · cited by 1,411
- Set.uIccstatement · cited by 393
- AbsolutelyContinuousOnIntervalstatement and proof · cited by 31
- UniformContinuousOn.continuousOnproof · cited by 3
- AbsolutelyContinuousOnInterval.uniformContinuousOnproof · cited by 2
Cited by2
Results whose statement or proof uses this declaration.
- AbsolutelyContinuousOnInterval.exists_boundproof · cited by 1
- AbsolutelyContinuousOnInterval.integral_mul_deriv_eq_deriv_mulproof · cited by 1