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

1.1given

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.

2.1step 1.1

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.

3.1step 1.1step 2.1∎

Steps 1.1 and 2.1 prove the example.

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.