DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-04

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.

Atoms and continuity points of a law

Definition

Let $X$ be a real random variable with law $P_{X}$ and cumulative distribution function $F_{X}$ .

A point $x \in R$ is an atom of the law of $X$ when it is an atom of the Borel measure $P_{X}$ , equivalently when $P_{X} ({x}) > 0.$
A point $x \in R$ is a continuity point of the law of $X$ when $F_{X}$ is continuous at $x$ .

The atom language belongs to the measure An atom of a measure on $R$ , while the continuity-point language belongs to the distribution function Cumulative distribution function of a real random variable.

Depends on

Used by

A distribution function need not have a density Counterexample
Discrete, continuous, and mixed distribution functions Example

Dependency tree · two levels

5 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.

Sources

J. R. Norris, Probability and Measure, Section 2.3 (standard reference, not scraped)