ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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 BV function can fail continuity at one point and still be differentiable almost everywhere

Example

Let

$H (x) : = {\begin{cases} 0, & 0 \leq x < 1 / 2, \\ 1, & 1 / 2 \leq x \leq 1. \end{cases}$

Then $H$ has bounded variation, is discontinuous at $1 / 2$ , and is differentiable almost everywhere with derivative $0$ .

Facts & Assumptions

Given: The step function $H$ above.

[A1]

The symbols are those of the statement.

Verification

technique · direct

1.1

For every partition of $[0, 1]$ , all endpoint increments vanish except possibly the one crossing $1 / 2$ , and that increment has absolute value $1$ . Hence the total variation of $H$ is $1$ , so $H$ is of bounded variation.

given

2.1

On each side of $1 / 2$ the function is locally constant, so $H^{'} (x) = 0$ for every $x \neq 1 / 2$ . Thus $H$ is differentiable almost everywhere. This is exactly the phenomenon asserted abstractly by Every function of bounded variation is differentiable almost everywhere.

step 1.1

3.1

Steps 1.1 and 2.1 prove the example.

step 1.1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

19 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.