Real_Function_is_Convex_iff_Derivative_is_Increasing

⚠️ Retired — specification defect

The Lean statement below does not encode the problem shown on this page, so its Disproved status carries no information about that problem. Do not import this node or use it as a dependency.

A differentiable function on a convex set is convex iff its derivative is non-decreasing.

Why this node was retired

The posted statement is

import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Calculus.MeanValue

theorem Real_Function_is_Convex_iff_Derivative_is_Increasing (f : ℝ → ℝ) (s : Set ℝ) (hs : Convex ℝ s) (hf : DifferentiableOn ℝ f s) : ConvexOn ℝ s f ↔ MonotoneOn (deriv f) s := by sorry

Differentiability within a closed convex set does not imply existence of the ordinary two-sided derivative at its boundary; totalized deriv can return zero there.

The recorded counterexample refutes the statement as encoded. It says nothing about the problem shown above, which is a different proposition.

Proposed corrected statement

Take an open interval and ordinary differentiability at every point, or explicitly assume DifferentiableAt f x for all x in the convex set before concluding monotonicity of deriv f. A within-derivative variant needs its own precise boundary convention.

Diagnosis and correction from the public Prove2Me statement audit (wamlat/prove2me-errors). The correction is natural-language mathematics and is not Lean-verified — it is a specification for a corrected node, not a drop-in replacement. No corrected replacement node exists yet.