Theorems · Theorem · real analysis

ConvexOn.slope_le_leftDeriv

∀ {S : Set ℝ} {f : ℝ → ℝ} {x y : ℝ},
  ConvexOn ℝ S f →
    x ∈ S → y ∈ S → x < y → DifferentiableWithinAt ℝ f (Set.Iio y) y → slope f x y ≤ derivWithin f (Set.Iio y) y

Reformulation of ConvexOn.slope_le_of_hasDerivWithinAt_Iio using derivWithin.

Defined in: Mathlib.Analysis.Convex.Deriv
Cited by: 1 results in Mathlib
Foundations: Depth 171 from the axioms · uses propext, Classical.choice, Quot.sound

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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.Iiostatement and proof · cited by 1,166
DifferentiableWithinAtstatement and proof · cited by 453
derivWithinstatement · cited by 258
ConvexOnstatement and proof · cited by 232
slopestatement · cited by 147
DifferentiableWithinAt.hasDerivWithinAtproof · cited by 85
ConvexOn.slope_le_of_hasDerivWithinAt_Iioproof · cited by 5

Cited by1

Results whose statement or proof uses this declaration.

ConvexOn.slope_le_leftDeriv_of_mem_interiorproof · cited by 1