TheoremStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-08-02

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A convex real function is Lipschitz on every closed bounded subinterval of the interior of its domain, hence continuous throughout the interior

Statement

Let $f : I \to R$ be convex and let $[a, b] \subseteq int I$ . Then there is $M \geq 0$ such that $∣ f (y) - f (x) ∣ \leq M ∣ y - x ∣$ for all $x, y \in [a, b]$ . Thus $f$ is Lipschitz on $[a, b]$ (Lipschitz map, $α$ -Hölder map for rational $0 < α \leq 1$ , and contraction) and is continuous at every point of $int I$ (Continuity of $f : A \to R$ at a point of $A$ and on $A$ : the $ε$ - $δ$ condition, its agreement with $\lim_{x \to c} f (x) = f (c)$ at a limit point, and continuity at an isolated point).

Facts & Assumptions

Given: A convex function $f : I \to R$ and $[a, b] \subseteq int I$ .

[L1]

For a convex function and $u < v < w$ , the three secant slopes satisfy $s (u, v) \leq s (u, w) \leq s (v, w)$ (For a convex function and $x < y < z$ , the three secant slopes satisfy $s (x, y) \leq s (x, z) \leq s (y, z)$ ).

[L2]

A function is Lipschitz with constant $L$ when $d (f (x), f (y)) \leq L d (x, y)$ for all points in its domain (Lipschitz map, $α$ -Hölder map for rational $0 < α \leq 1$ , and contraction).

Proof

technique · direct

1.1L1

Choose $u, v \in I$ with $u < a \leq b < v$ ; then for $a \leq x < y \leq b$ , two applications of the three-slope inequality give $s (u, a) \leq s (x, y) \leq s (b, v)$ .

2.1step 1.1L2algebra

With $M : = \max {∣ s (u, a) ∣, ∣ s (b, v) ∣}$ , step 1.1 yields $∣ f (y) - f (x) ∣ \leq M ∣ y - x ∣$ for $x < y$ ; symmetry gives the same estimate for all $x, y \in [a, b]$ , which is the Lipschitz condition.

3.1step 1.1step 2.1∎

Given $c \in int I$ , choose such an interval $[a, b]$ containing $c$ in its interior; the estimate in step 2.1 gives the $ε$ -- $δ$ condition at $c$ by taking $δ = ε / M$ when $M > 0$ , and is immediate when $M = 0$ .

Depends on

Dependency tree · 2 levels

Sources

R. Gardner, Convex Functions, Notes 6.6