Theorems · Theorem · real analysis

HasDerivWithinAt.nonneg_of_monotoneOn

∀ {𝕜 : Type u} [inst : NontriviallyNormedField 𝕜] {x : 𝕜} {s : Set 𝕜} [inst_1 : LinearOrder 𝕜] [IsStrictOrderedRing 𝕜]
  [OrderTopology 𝕜] {g : 𝕜 → 𝕜} {g' : 𝕜},
  AccPt x (Filter.principal s) → HasDerivWithinAt g g' s x → MonotoneOn g s → 0 ≤ g'

If a monotone function has a derivative within a set at a non-isolated point, then this derivative is nonnegative.

Defined in

Mathlib.Analysis.Calculus.Deriv.Slope

Cited by

4 results in Mathlib

Foundations

Depth 169 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldLinearOrderIsStrictOrderedRingOrderTopology