DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-01

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.

Translation of a function on $R^{n}$

Definition

Let $h \in R^{n}$ and let $f : R^{n} \to C$ be a function. The translate of $f$ by $h$ is the function $τ_{h} f$ defined by

(τ_{h} f) (x) : = f (x - h) (x \in R^{n}) .

This fixes the sign convention used throughout the page: translating by $h$ shifts the graph of $f$ in the positive $h$ -direction.

Used by

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

Richard L. Wheeden and Antoni Zygmund, Measure and Integral: An Introduction to Real Analysis (standard reference, not scraped)