False statementConstruction: AI-adaptedVerification: AI-adaptedprecheck passaudited 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.

FALSE: translation is continuous in $L^{\infty}$

Statement

False claim. Translation is continuous in $L^{\infty} (R^{n})$ .

Facts & Assumptions

Given: The indicator of the unit interval in one dimension.

[L1]

The finite- $p$ translation theorem excludes $p = \infty$ ( $∥ τ_{h} f - f ∥_{p} \to 0$ in $L^{p} (R^{n})$ as $h \to 0$ , for $1 \leq p < \infty$ ).

Refutation

technique · direct

1.1

Let $f : = 1_{[0, 1]}$ on $R$ . For every $h \neq 0$ , the [L1, given, algebra] functions $f$ and $τ_{h} f$ differ by $1$ on a set of positive measure: for $0 < h < 1$ , for instance, on $(1, 1 + h]$ one has $τ_{h} f = 1$ and $f = 0$ .

L1givenalgebra

2.1

Therefore [step 1.1, algebra]

∥ τ_{h} f - f ∥_{\infty} = 1

for every $h \neq 0$ , so the norm does not tend to $0$ as $h \to 0$ .

step 1.1algebra

3.1

Hence translation is not continuous in $L^{\infty}$ , which is exactly why [L1, step 2.1] [L1] stops at $p < \infty$ .

L1step 2.1∎

Depends on

$\|\tau_h f - f\|_p \to 0$ in $L^p(\mathbb{R}^n)$ as $h \to 0$, for $1 \le p < \infty$

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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

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

FALSE: translation is continuous in L∞L∞