Theorems · Theorem · real analysis
lipschitzWith_of_nnnorm_deriv_le
∀ {𝕜 : Type u_3} {G : Type u_4} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup G] [inst_2 : NormedSpace 𝕜 G] {f : 𝕜 → G}
{C : NNReal}, Differentiable 𝕜 f → (∀ (x : 𝕜), ‖deriv f x‖₊ ≤ C) → LipschitzWith C fThe mean value theorem set in dimension 1: if the derivative of a function is bounded by C,
then the function is C-Lipschitz. Version with deriv and LipschitzWith.
- Defined in
- Mathlib.Analysis.Calculus.MeanValue
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 195 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites12
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- NormedAddCommGroupstatement and proof · cited by 15,752
- NormedSpacestatement and proof · cited by 12,499
- NNRealstatement and proof · cited by 4,310
- Set.univproof · cited by 3,945
- RCLikestatement and proof · cited by 2,829
- NNNorm.nnnormstatement and proof · cited by 952
- derivstatement and proof · cited by 676
- LipschitzWithstatement · cited by 316
- Differentiablestatement and proof · cited by 298
- convex_univproof · cited by 53
- lipschitzOnWith_univproof · cited by 11
- Convex.lipschitzOnWith_of_nnnorm_deriv_leproof · cited by 1
Cited by4
Results whose statement or proof uses this declaration.
- Real.lipschitzWith_cosproof · cited by 1
- Real.lipschitzWith_one_mulExpNegMulSqproof · cited by 1
- Real.lipschitzWith_sinproof · cited by 1
- lipschitzWith_circleMapproof · cited by 0