Theorems · Theorem · real analysis

ConvexOn.strictMonoOn

∀ {𝕜 : Type u_1} [inst : Field 𝕜] [inst_1 : LinearOrder 𝕜] [IsStrictOrderedRing 𝕜] {s : Set 𝕜} {f : 𝕜 → 𝕜},
  ConvexOn 𝕜 s f → ∀ {x y : 𝕜}, x ∈ s → x < y → f x < f y → StrictMonoOn f (s ∩ Set.Ici y)

If f is convex on a set s in a linearly ordered field, and f x < f y for two points x < y in s, then f is strictly monotone on s ∩ [y, ∞).

Defined in

Mathlib.Analysis.Convex.Slope

Cited by

3 results in Mathlib

Foundations

Depth 64 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FieldLinearOrderIsStrictOrderedRing