Theorems · Theorem · real analysis
ConvexOn.hasDerivWithinAt_sSup_slope_of_mem_interior
∀ {S : Set ℝ} {f : ℝ → ℝ} {x : ℝ},
ConvexOn ℝ S f → x ∈ interior S → HasDerivWithinAt f (sSup (slope f x '' {y | y ∈ S ∧ y < x})) (Set.Iio x) x- Defined in
- Mathlib.Analysis.Convex.Deriv
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 170 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites32
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
- Set.ofPredstatement and proof · cited by 6,101
- Set.imagestatement and proof · cited by 5,609
- nhdsproof · cited by 5,554
- Filter.Tendstoproof · cited by 3,814
- LE.le.transproof · cited by 3,151
- LT.lt.leproof · cited by 2,189
- nhdsWithinproof · cited by 1,912
- Set.Iooproof · cited by 1,214
- Set.Iiostatement and proof · cited by 1,166
- SupSet.sSupstatement and proof · cited by 954
Cited by2
Results whose statement or proof uses this declaration.
- ConvexOn.leftDeriv_eq_sSup_slope_of_mem_interiorproof · cited by 2
- ConvexOn.differentiableWithinAt_Iio_of_mem_interiorproof · cited by 2