TheoremStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02

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

Statement

Let $f:I\to\mathbb R$ be convex and let $[a,b]\subseteq\operatorname{int}I$ . Then there is $M\ge0$ such that $|f(y)-f(x)|\le M|y-x|$ for all $x,y\in[a,b]$ . Thus $f$ is Lipschitz on $[a,b]$ (Lipschitz map, $\alpha$ -Hölder map for rational $0 < \alpha \le 1$ , and contraction) and is continuous at every point of $\operatorname{int}I$ (Continuity of $f : A \to \mathbb{R}$ at a point of $A$ and on $A$ : the $\varepsilon$ - $\delta$ 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 $f:I\to\mathbb R$ and $[a,b]\subseteq\operatorname{int}I$ .

[L1]

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

[L2]

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

Proof

technique · direct

1.1

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

2.1

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

step 1.1L2algebra

3.1

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

step 1.1step 2.1∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 50 results over 12 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

R. Gardner, Convex Functions, Notes 6.6 (standard reference, not scraped)