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

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A convex function on an open interval is differentiable except at at most countably many points

Statement

If $f:I\to\mathbb R$ is convex on an open interval, then the set of points at which $f$ is not differentiable is at most countable.

Facts & Assumptions

Given: A convex $f:I\to\mathbb R$ on an open interval.

[L1]

A convex function on an open interval has finite left and right derivatives everywhere, and for $u<v$ they satisfy the ordered one-sided-derivative chain (A convex function on an open interval has finite left and right derivatives everywhere, with $f'_-(u)\le f'_+(u)\le (f(v)-f(u))/(v-u)\le f'_-(v)\le f'_+(v)$ for $u<v$ ).

[L2]

The discontinuity set of a monotone real function on an interval is at most countable (Froda's theorem: the set of discontinuities of a monotone function on an interval is at most countable, the injection into $\mathbb{N}$ being built from one fixed enumeration of the rationals by least index, so no choice principle is used).

[L3]

A convex real function is continuous throughout an open interval on which it is convex (A convex real function is Lipschitz on every closed bounded subinterval of the interior of its domain, hence continuous throughout the interior).

Proof

technique · direct

1.1

Put $g(c):=f'_+(c)$ . The order chain in [L1] gives $g(u)\le g(v)$ whenever $u<v$ , so $g$ is nondecreasing.

1.2

For $u<c$ , [L1] gives $g(u)\le f'_-(c)$ . Conversely, if $a<u<c$ , [L1] gives $s(a,u)\le f'_-(u)\le g(u)$ . Letting $u\to c^-$ , continuity from [L3] gives $\lim_{u\to c^-}s(a,u)=s(a,c)$ ; then letting $a\to c^-$ gives $\lim_{u\to c^-}g(u)=f'_-(c)$ . Since $g(c)=f'_+(c)$ , nondifferentiability of $f$ at $c$ makes $g$ discontinuous there.

L1L3algebra

2.1

Froda's theorem makes the discontinuity set of $g$ at most countable, and step 1.2 places every nondifferentiability point of $f$ in that set.

L2step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 81 results over 17 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)