Theorems · Theorem · real analysis
ContDiffWithinAt.exists_lipschitzOnWith
∀ {E : Type u_4} {F : Type u_5} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [inst_2 : NormedAddCommGroup F]
[inst_3 : NormedSpace ℝ F] {f : E → F} {s : Set E} {x : E},
ContDiffWithinAt ℝ 1 f s x → Convex ℝ s → ∃ K, ∃ t ∈ nhdsWithin x s, LipschitzOnWith K f tIf f is C^1 within a convex set s at x, then it is Lipschitz on a neighborhood of x
within s.
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 196 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites27
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
- Realstatement and proof · cited by 25,697
- NormedAddCommGroupstatement and proof · cited by 15,752
- NormedSpacestatement and proof · cited by 12,499
- Filterstatement and proof · cited by 8,121
- ENatstatement · cited by 4,985
- NNRealstatement and proof · cited by 4,310
- WithTopstatement · cited by 3,754
- nhdsWithinstatement and proof · cited by 1,912
- le_rflproof · cited by 1,558
- Metric.ballproof · cited by 735
- FormalMultilinearSeriesproof · cited by 615
Cited by2
Results whose statement or proof uses this declaration.
- ContDiffOn.dimH_image_leproof · cited by 2
- ContDiffOn.locallyLipschitzOnproof · cited by 1